Macintosh OSX Maintenance

Backup
Backup is pretty essential before you start monkeying about with any other tool, I've already written a lens about backing up your mac. Basically if you've got Leopard, do yourself a favour and get an external hard drive and use Time Machine.

Repair permissions
This is contentious (as are many Mac maintenance topics), some people maintain you never need to repair permissions. See Macattoneys for a full discussion
I think this is wrong basically, permissions repair is free, easy and can fix some slightly strange errors. I've personally had experience of a Mac that refused to wake up from sleep which was fixed by permissions repair.
To Repair permissions just start up Disk Utility and click on the repair permissions button. You can automate this with Cocktail or Onyx. Cocktail costs money ($15 for a single user license) but is the better tool, Onyx is free.

Drive Repair
Similarly you can repair drives using Disk Utility. You'll need to boot from the system disks as a repair won't work on a boot disk.
If the repair fails then your best bet is DiskWarrior.

Routine Maintenance/ Clear Caches
You no longer need to run the log rotation jobs in 10.5 (Leopard), as the machine runs them even if you don't leave your machine on overnight. With previous versions of the OS you needed to run something like Cocktail or MacJanitor to run the log jobs.
Clearing caches can be done for free by Cache out X

Defragging
Again, prone to controversy. The truth of the matter is slightly involved but the Mac OS will automatically defrag files under 20MB, however if you have files over this size they are prone to fragmentation. Also the OS does nothing about directory fragmentation (spaces between files). You may not ever notice the problem however if you keep 20% of your disk drive free and you never use bootcamp to install Windows.
However I've run across this problem, where you need to defrag to partition the drive.
The best defragger is the stand alone iDefrag, which is particularly OSX savvy and keeps critical files where the OS needs them. You can use Tech Tools and Drive Genius to defrag a drive too, but they are less powerful. Note iDefrag needs you to boot from a disk you create using Coriolis Disk Maker that comes with the program.

Partiton Drives
Partitioning a drive is new in Leopard and has some limitations, however the existing functionality can mean you don't need to buy third party tools if you just want to create one or more new partitions.

Boot from the system disk and launch Disk Utility , select your hard drive on the left and click on the partition tab and you will get a Window with the current partitions. You can drag these around by using the resize control at the bottom of the partitions or click on the '+' to create a new partition. You'll need to name your new partitions. Click on Apply and your new partition will be created.
Note that the only thing you can't do with the partitioning software is move the partitions, so if Partition A ends at 50Gb and Partition B starts there, then you can't decrease Partition A and move partition B into the space , otherwise it leaves dead space on the drive. To move the start point of any partition you need third party software such as iPartiton, there's a nice review on macosxhints . There's also my experience of trying to reformat without it!
You an also partition using TechTools Pro and Drive Genius and an open source tool called gparted. iPartition is the only one that will re-partition bootcamp partitions on the fly.