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.
Retail installer discs icons are included where possible, if you have an older machine that is compatible.
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)
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) (all known as Jaguar). 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. 10.2.8 is the end of the line for all beige G3 and black Powerbook G3 with 9 pin serial ports not listed.
Mac OS X 10.3.9 from 10.3.0 through 10.3.8 . (all USB G3s except some Powerbook Bronze keyboard) (all known as Panther). Check this tip for updating to 10.3 and later.
10.3.9 is the end of the line for all USB iMacs, Powerbooks, and iBooks without Firewire
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**) (all known as Tiger).
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) (all known as Leopard). 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***) (all known as Snow Leopard). 10.6.8 is the last operating system to have built-in Rosetta support for PowerPC applications. The implications of that are on this tip. See this tip for updating to 10.6 or later.
It is the last Mac operating system to be made available on DVD.
Mac OS X 10.7.5 from 10.7.0 through 10.7.4 (Intel Core2Duo, Xeon, Core i processor Macs) (All known as Lion)
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. A supplemental update to 10.7.5 is also available, which is mentioned on the tip. It is the latest operating system to be made available on physical media. A Flash Drive was available during its reign at the Apple Store.
Mac OS X 10.8.5 from 10.8 through 10.8.4 (Mountain Lion) 10.8 was released July 25, 2012 and has this tip discussing its compatibility. A supplemental update to 10.8.2 is also mentioned on the tip. Apple is offering a paid update from 10.6.8, and any version of 10.7 to 10.8 through the Mac App Store.
Mac OS X 10.9 Mavericks, breaks the tradition, and is available free (subject certain license restrictions) for anyone from 10.6.8 through 10.8.5 as long as they meet the system requirements for 10.8, but it also may have issues with compatibility that have not yet been addressed, but have been for 10.8, simply because it just came out on October 22, 2013. 10.9.2 is strongly recommended for anyone upgrading to 10.9, as there is a critical security flaw in 10.9. 10.9.3, 10.9.4 and 10.9.5 are also available as free updates to 10.9. This user tip discusses the uniqueness of 10.9. Minimum hardware requirements are the same as 10.8.
Mac OS X 10.10 Yosemite is very similar to Mavericks, except with enhanced features for newer Macs. It has 10.10.1, 10.10.2, 10.10.3, 10.10.4, and 10.10.5 as available updates. Minimum hardware requirements are the same as 10.9 and 10.8. See this tip.
Mac OS X 10.11 El Capitan is also similar to Mavericks. See this tip.
Outside of Mavericks, and Yosemite, it means you can't directly update free 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.
And the Mavericks and Yosemite "free" update was actually more costly than the other intermediate updates already mentioned, as the full release operating systems frequently had interface changes and significant changes that altered how certain applications and drivers worked together. Usually within a major release application and driver compatibility remained consistent.
Prior to Lion the only other exception existed for Macs purchased within three weeks of a full operating system release, when certain Up To Date installation discs, also known as DropIn discs were made available to help those update who bought a new Mac just as the operating system transitioned. Those discs were only allowed to be used by certain Macs within that time frame. DropIn discs were included with new Macs after the release. Either way, DropIn, or UpToDate Macs had the advantage of being able to run both the new and old operating system. Up To Date discs could be requested from Apple for the Macs that qualified. After that date a hardware change in a Mac model would almost always mean the prior operating system would no longer be useable on the new Mac. It has been found though you can run 10.6 Server on Macs that shipped new with 10.7 or later using virtualization.
All of this does not mean that some operating systems can't be skipped when upgrading from one retail to another and purchasing that new retail release. 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