These make a complete, "bootable" copy of your entire system on an external disk/partition, a second internal disk/partition, or a partition of your internal disk.

Advantages

1. •If your internal HD fails, you can boot and run from the clone immediately.  Your Mac may run a bit slower, but it will run, and contain everything that was on your internal HD at the time the clone was made or last updated.  (But of course, if something else critical fails, this won't work.)
   

2. •You can test whether it will run, just by booting-up from it (but of course you can't be positive that everything is ok without actually running and testing everything).
   

3. •If it's on an external drive, you can easily take it off-site.
   

4. 
   

Disadvantages

1. •Making an entire clone takes quite a while.  Most of the cloning apps have an update feature, but even that takes a long time, as they must examine everything on your system to see what's changed and needs to be backed-up.  Since this takes lots of time and CPU, it's usually not practical to do this more than once a day.
   

2. •Normally, it only contains a copy of what was on your internal HD when the clone was made or last updated.
   

3. •Some do have a feature that allows it to retain the previous copy of items that have been changed or deleted, in the fashion of an archive, but of course that has the same disadvantages as an archive.  And you have to navigate in each backup separately, unlike the Time Machine display.
   

4. •Clones only back up a single volume (disk or partition) at a time, usually to a separate destination.  If you have multiple volumes, you must run the clone multiple times.  But you can set up one or more such tasks to run at different times.
   

5. •Cloning via a network is problematical.  Some cloning apps won't work over a network at all, and some that do don't recommend it.  Some suggest cloning to a disk image, but since you can't boot from a disk image, you'd have to restore it somewhere first.  Worse, if the clone's on a network, you can't boot from it directly;  you must copy it somewhere that is bootable.
   

6. •Once you start actually running from a clone, it becomes your original, and it needs to be backed-up.  If you don't do that, you no longer have backups.

Similar to an archive, Time Machine keeps copies of everything currently on your system, plus previously changed/deleted items, on an external disk, Time Capsule (or USB drive connected to one), internal disk, or shared drive on another Mac on the same local network.

Advantages

1. •Like many Archive-type apps, it first copies everything on your system, then does incremental backups (hourly, as long as your Mac is awake and the destination is available) of additions and changes.  But Time Machine's magic is, each backup is, in effect, a full one:   a complete copy of everything on your system at the time of the backup.
   

2. •It uses an internal OSX log of what's changed to quickly determine what to copy, so most users can let it do its hourly incremental backups without much effect on system performance.  This means you have a much better chance to recover an item that was changed or deleted in error, or somehow got corrupted.  That also means it's much quicker to see what's new or changed and needs to be backed-up.
   

3. •Recovery of individual items is quite easy, via the Time Machine interface, since it looks just like the Finder, or one of the apps that have special handling for Time Machine.   You can browse your backups just as your current data, and see "snapshots" of the entire contents at the time of each backup.  You don't have to find and mount media, or dig through many files to find what you're looking for.
   

4. •You can also recover your entire system (OSX, apps, settings, users, data, etc.) to the exact state it was in at the time of any backup, even it that's a previous version of OSX. 
   

5. •Effective with Lion, Time Machine also makes hourly Local Snapshots on the internal HD of laptops, and keeps them for 24 hours, then one per day for a week.  So if you're away from your normal backups, and have a problem, you have a good chance to recover an earlier version of a file that was changed or deleted in error, or got corrupted.  These don't really take up space, as they'll be deleted automatically as necessary to keep your disk under 80% full.
   

6. •Time Machine manages its space for you, automatically.  When your backup disk gets near full, Time Machine will delete your oldest backup(s) to make room for new ones.  But it will never delete its copy of anything that's still on your internal HD, or was there at the time of any remaining backup.  So all that's actually deleted are copies of items that were changed or deleted long ago.
   

7. •Time Machine can back up multiple volumes (disks and/or partitions) to a single volume, in the same backup process. 
   

8. •Effective with Mountain Lion, you can specify two or more backup drives (local and/or network); Time Machine will back up to them alternately, when available.
   

9. •Time Machine examines each file it's backing-up; if it's incomplete or corrupted, Time Machine may detect that and fail, with a message telling you what file it is.  That way, you can fix it immediately, rather than days, weeks, or months later when you try to use it.
   

10. •Effective with Lion, local backups can be encrypted;  effective with Mountain Lion, network backups can be encrypted, too.
    

11. If you're interested in the "works," see  How Time Machine Works its Magic.
    

Disadvantages

1. •It’s not bootable.  If your internal HD fails, you can't boot directly from your Time Machine backups.  You must restore them, either to your repaired/replaced internal HD or an external disk.   This is a fairly simple, but of course lengthy, procedure.  You can also transfer the apps, user accounts, and data to another disk or Mac, via Setup Assistant or Migration Assistant.  See How do I set up a new Mac from an old one, its backups, or a PC? for details.
   

2. However, if you're running Lion 10.7.2 or above, and backing-up to a directly-connected external HD, there's probably a copy of your Recovery HD on the Time Machine drive, so if your internal HD fails, you can start from that.  See Using the Recovery HD.
   

3. •Time Machine doesn't keep its copies of changed/deleted items forever, and you're usually not notified when it deletes them.
   

4. •It can back up to many network locations, but not from any.
   

5. •Time Machine is relatively complex internally, and while it’s been around for a while, there are still some things it doesn’t do (like back up your photos while iPhoto is open). 

For most non-professional users, Time Machine is simple, workable, and maintenance-free.  See the Time Machine Tutorial and/or browse the Time Machine FAQ for more information. But it does have its disadvantages.  More importantly, all hardware fails, sooner or later.  And those of us who haven't erased the wrong drive yet probably will, one of these days.

That's why many folks use both Time Machine and a bootable clone, to have two separate, independent backups, with the advantages of both.  If one fails, the other remains.  If there's room, these can be in separate partitions of the same external drive, but it's safer to have them on separate drives, so if either drive fails (they all do, sooner or later), you still have the other one.

Bootable clones are usually made with either the CarbonCopyCloner or SuperDuper! apps.  CCC is usually $40 U.S., with a 30-day free trial;  SD has a free version, but you need the paid one (about $30 U.S.) to do updates instead of full replacements, or scheduling. 

For one-time use (full replacement only;  no updating), the Restore tab of Disk Utility will work, too.  See #7 in Using Disk Utility for details.

Some other options are: Data Backup, Deja Vu, Synk Pro, ChronoSync, and Tri-Backup

As great as external drives are, they may not protect you from fire, flood, theft, or direct lightning strike on your power lines.  So it's an excellent idea to get something off-site, to your safe deposit box, workplace, relative's house, or other secure location.

There are many ways to do that, depending on how much data you have, how often it changes, how valuable it is, and your level of paranoia.

One of the the best strategies is to follow the above recommendation to use Time Machine locally, but with a pair of portable externals for a "bootable clone."   Use one of them for a week or so, then take it off-site and swap with the other.  That way, no matter what happens, there's always a copy in the secure location.

There are other options, instead of the dual drives, or in addition to them.  Your off-site backups don't necessarily have to be full backups, but could be just copies of critical information.

You can use a 3rd-party online backup service such as DropBox, Mozy or Carbonite.  Trying to back up your entire system that way may not be workable, but smaller amounts of specific data may be. 

You can also copy data to CDs or DVDs and take them off-site.   Re-copy them every year or two, as their longevity is questionable, and be sure to keep them in a cool, dry place, out of direct sunlight.

Backup strategies are not a "One Size Fits All" sort of thing.  What's best varies by situation, budget,  and preference.

Just as an example, I keep at least two sets of full Time Machine backups;  plus a CarbonCopyCloner clone (updated automatically early each morning, while I'm snoozing) locally;  plus another on a portable external HD in my safe deposit box; and some things online as well.  Probably overkill, but as many of us have learned (the hard way) over the years, backups are one area where Paranoia is Prudent!

These copy specific files and folders, or in some cases, your entire system.   Usually, the first backup is a full copy of everything; subsequently, they're "incremental," copying only what's changed.

Most of these will copy to an external disk; some can go to a network locations,  some to CDs/DVDs, or even tape. 

Advantages

1. •They're usually fairly simple and reliable.  If the increments are on separate media, they can be taken off-site easily.
   

Disadvantages

1. •Most have to examine everything to determine what's changed and needs to be backed-up.  This takes considerable time and lots of CPU.  If an entire system is being backed-up, it's usually not practical to do this more than once, or perhaps twice, a day.
   

2. •Restoring an individual item means you have to find the media and/or file it's on.  You may have to dig through many incremental backups to find what you're looking for.
   

3. •Restoring an entire system (or large folder) usually means you have to restore the most recent Full backup, then each of the increments, in the proper order.  This can get very tedious and error-prone.
   

4. •You have to manage the backups yourself.  If they're on an external disk, sooner or later it will get full, and you have to do something, like figure out what to delete.   If they're on removable media, you have to store them somewhere appropriate and keep track of them.  In some cases, if you lose one in the "string" (or it can't be read), you've lost most of your backup.