Some people may not be aware, but "download" means to retrieve off a network server. In the discussion below the network refers to the internet, and the server in question is Apple's server. Before you download be sure to backup your data. Link for backing up only refers to Mac OS X applications. If you need help finding
how to backup other operating systems, be sure to ask in the appropriate forum.
If the source is Apple http://support.apple.com/downloads/ and http://www.info.apple.com/support/oldersoftwarelist.html you can download updates to your operating system, assuming you meet the requirements given on the download page. Any source outside of Apple is suspect, and you should inquire on Apple Discussions before attempting to download such updates to determine if your update is able to be applied to your system.
The following are the latest operating systems systems available for free download:
6.0.8 from any previous OS (applies to 128k, 512k, 512ke),
7.5.5 from any previous OS (applies to original II, Plus, LC, Portable, and Powerbook 100, SE/30, Classic) if your machine has the proper enablers.
7.6.1 from 7.6 (applies to all 68030 Motorola)
8.1 from 8.0 (applies to all 68040 Motorola)
8.6 from 8.5 and 8.5.1
9.2.2 (applies to select G4s, G5s, all G3s, except original 250 Mhz Powerbook from November 10, 1997 through March 14, 1998) from 9.0, 9.0.2, 9.0.4, 9.1 (applies to all PowerPC pre-G3 except some 54xx, 53xx, 52xx, 64xx, 63xx, 62xx series, and original 250 Mhz Powerbook G3), 9.2.1. Some G4s did not come prebundled with 9.2.2, and could be upgraded to 9.2.2, the ones which didn't were released after July 2002. Of those that had 9.2.2 included, only a few could boot into 9. No G5 could boot into 9, but all could run Classic. 9 updates which used to be available on Apple's support website can be found elsewhere according to this tip.
Mac OS X 10.0.4 from 10.0.0, 10.0.1, 10.0.2, 10.0.3 (all G3s, except original Powerbook G3/250 which couldn't install any Mac OS X, and iMac Summer 2001 which could only install 10.2 or later in terms of Mac OS X)
Mac OS X 10.1.5 from 10.1.1, 10.1.2, 10.1.3, 10.1.4 (all G3s, except original Powerbook G3/250 which couldn't install any Mac OS X, and iMac Summer 2001 which could only install 10.2 or later in Mac OS X version)
Mac OS X 10.2.8 from 10.2.0 through 10.2.7. (all G3s, except original Powerbook G3/250 which couldn't install any Mac OS X). Be sure to upgrade your firmware before installing any later operating system if your Mac supports this operating system and has not been updated yet, and check for these other tidbits for installing 10.2 or later.
Mac OS X 10.3.9 from 10.3.0 through 10.3.8. (all USB G3s except some Powerbook Bronze keyboard). Check this tip for updating to 10.3 and later.
Mac OS X 10.4.11 from 10.4.0 through 10.4.10 (all Firewire built-in with Firewire DVD* or built-in DVD drive PowerPC Macs, and pre-October 26, 2007 Intel Macs**).
Check this tip for updating to 10.4 and later.
Mac OS X 10.5.8 from 10.5.0 through 10.5.7 (all 867 Mhz or faster built-in processor PowerPC, and Intels prior to August 28, 2009). Check this tip for updating to 10.5 and later.
Mac OS X 10.6.8 from 10.6.0 through 10.6.7 (all Intel Macs***). Check this tip for updating to 10.6 or later.
Mac OS X 10.7.5 from 10.7.0 through 10.7.4 (Intel Core2Duo, Xeon, Core i processor Macs)
If the right hand side operating systems on the rows above do not list your operating system, you will need to buy one of those intermediate operating systems retail, assuming your machine allows that operating system by specs. Note a paid download from the Mac App Store and a USB Flash drive directly from 10.6.6, 10.6,7, or 10.6.8 is possible. Check this tip for more on updating to 10.7 or later.
Mac OS X 10.8.2 from 10.8 through 10.8.1. 10.8 was released July 25, 2012 and has this tip discussing its compatibility. Apple is offering a paid update from 10.6.8, and any version of 10.7 to 10.8 through the Mac App Store.
That means you can't directly update from 10.5.8 to 10.6.1,
You can't directly update from 10.4.11 to 10.5.8
etc without purchasing the intermediate retail update on a CD or DVD or floppy. This does not mean that some operating systems can't be skipped when upgrading from one retail to another. A person buying 10.6 retail can install on top of 10.4.4 or later, as long as it is an Intel Mac without having to purchase 10.5. A person buying 10.7 USB flash drive can install on an erased hard drive of any Intel Xeon, or Core2Duo or better Mac that shipped with 10.4.4 or later and has 2 GB of RAM, though has to have 10.6.6 or later to upgrade and install to 10.7. The Mac App Store is another means of obtaining 10.7, and that's only available in 10.6.6 and later. If uncertain if a possible skip is possible, ask on the discussion forum relevant to the upgrade.
* Blue and White G3s, and PCI PowerMac G4s could not boot off an external DVD drive, but could have these operating systems installed if they had a built-in DVD drive, or you are able to find the limited edition Media Exchange Program CD-ROM version of 10.4.
** Intel Macs could only have 10.4 installed from the prebundled discs that came with them, and no retail 10.4 would install.
*** March 15, 2010 to July 19th 2011 Macs could not install the retail 10.6, but could install Lion retail.
July 20, 2011 and later Macs could only install 10.7 or later retail releases. The date of the Mac is based on when the hardware was refreshed. The tip next to the 10.6 operating systems will tell you which Macs can install 10.6, and which can only install 10.7.
Note: Additional limitations like above are indicated in Apple article: http://support.apple.com/kb/HT2186