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EveMon is primarily a skill planning tool, but has some other features as well. It's almost a must have and is widely used.
Eve Fitting Tool (EFT) is one of the two most used fitting tools.
Link: https://forums.eveonline.com/default.aspx?g=posts&t=24359
All fits posted on this wiki are posted in EFT format and will import straight into the program (once it is on your clipboard)
Pyfa is the other widely used tool. It's Python based, which means it works with Linux based OSs
jEveAssets is an asset tracking program, which is especially useful if you have a lot of characters, or a lot of items (or both).
Evernus is primarily a trading tool, which tracks your buy and sell orders, assets and profits (or losses)
Link: http://evernus.com/
EveHQ is a swiss army knife application that does a little bit of everything
Link: http://evehq.co/
GARPA is a Goon made program for travel planning, both for Capitals and Sub Capitals.
Link: http://gts.goonfleet.com/gts-release-3100.exe
Info: https://www.reddit.com/r/Eve/comments/3lh1pl/garpa_31_liberated/
Another Goon tool for cloning your UI across multiple accounts.
Link: http://s3.amazonaws.com/garpa-distribution/GarpaUI2.zip (DIRECT DOWNLOAD)
Guide: http://s3.amazonaws.com/garpa-distribution/GarpaUI2-manual.pdf
This isn't an application, but it probably should be. It's used for planning how to build a POS