Game Editors
Here is a list of some 3rd party software that can be used to manipulate UFO:Enemy Unknown.
XCOM-specific tools (including TFTD)
XcomUtil, originally by Scott Jones and now being updated by BladeFireLight, can perform many useful fixes and changes to both EU and TFTD. It can also create hybrid games that mix the terrain from both games. Most changes are relatively easy using the XcuSetup.bat script. Others are harder and require you to edit text configuration files. The interface is currently being updated to a new, much improved version, making this easier to use and reliable, as well as powerful. It works with Steam distributions and DOSBox.
UFO Extender (by Seb76) can fix various bugs from the CE edition of the game. It also provides a large number of useful gameplay enhancements. UFO Extender allows you to play the game "as it should've been programmed originally". It also allows you to play alternate scenarios and rules for making the game harder (or easier). It works with XComUtil and the Steam distributions, but only for the Windows CE version. Works only with EU, not TFTD.
TFTD Extender (by Tycho/Morgan525) is the conversion of UFO Extender to Terror From the Deep. It provides a large number of the same enhancements as the original as well as correcting many ones specific to TFTD.
Bomb Bloke has a handy collection of tools on this site, for doing all sorts of customising. The Battlescape Editor is an excellent visual editor for adding, removing and changing units and items on a particular saved game, including changing the stats of weapons and other items. Also includes tools for modding uniforms, graphics/artwork, and other useful goodies.
UFO Classes (by Necuno) lets you manage the selection, deployment and training of soldiers via a concept of classes and levels, rather than having to micromanage soldiers based on all their specific stats. Works only with EU, not TFTD.
XCOMHack and other tools by Chris Voss (Hatfarm). XComHack is a VB program that can edit soldiers, bases, craft, diplomacy and global factors such as money. Pretty much everything that ClarkWehyr can do and easy to use. It is a Visual Basic Forms application and the source code is available to extend. This is probably the easiest to use saved game editor, as it uses a standard GUI familiar to most people. The TFTD version does not have as many features as the UFO version. There are also some more powerful utilities available here to edit & patch the UFO executable - UFO Mod and Patch Maker.
SuperMen is a simple utility that turns your soldiers to, well, supermen.
Older tools
UE Matt Mullen's UE editor, just for editing bases and money. A very simple DOS character mode "GUI". This also comes with source code and is written (nicely) in C. Nice.
XcomUtil also includes a utility called SDump. This is a hex dumper / patcher that is configured for XCom. Very technical and old skool but very powerful. Use with care and read the manual first. Probably only usable by old skool programmers. For many patches (eg craft, tank, weapon and alien stats) it will be easier just to modify the xcomutil.cfg file.
ClarkWehr editor. This is a soldier and base editor for XCOM and TFTD. Reportedly has some bugs. It has a GUI but it is character-mode DOS type of GUI. Also has command line switches so you can run it in batch mode. Lots of features including a Soldier Exporter (to spreadsheet).
There's at least one editor that allows you to change the weapons: XComEd.
You can find many X-COM game editors, cheats and mods over at StrategyCore's Files Section.
Useful Generic File Editors
MS-Edit MS-Edit is a simple text editor that can be used to manipulate binary data. While a very old program dating back to the days of MS-DOS, recent versions of Windows up to XP still come bundled this program. It offers a binary mode that, while very unconventional, can be used to manipulate binary data in files. With knowledge of the game file structures and the nuances of how MS-Edit works, you can go right to the source and exact changes as you see fit in practically any game file.
[HexIt] by Mikael Klasson is a free and powerful hex editor. Donation suggested if you use it a lot. It also has a decompiler view if you want to try to figure out what the executable is doing.
[Frhed] is a free, open source hex editor that works well.
Another VERY useful hex editing program is the Universal Game Editor (UGE). It is a hex editor which displays numbers in digits instead of hex codes. It uses templates that can store variables found in the game files, so that they can be easily edited (including variables with 1, 2, or 4 bytes, or text entries).
It can be found from a variety of locations using a search engine. The program is only about 400kb and is free to use (shareware with full capabilities, but has a reminder screen).
Mikael 'Fluff' Klasson makes HexIt, a very powerful hex editor that includes a dissassembler. Useful for guessing what the game engine is trying to do and deducing game mechanics (or bugs). HexIt is freeware, though there is a suggested donation if you find it useful.