Many mods don't work. At several points in the past, the game's patches have caused existing mods to stop functioning. "Active" mods, those still supported by their creators, get updated with new versions post-patch, but many of the older mods' creators have long since abandoned Civ5 modding, and so their creations were never fixed.
If you're downloading a mod through the in-game browser, you just shouldn't use anything that hasn't been updated recently. This isn't always easy to tell, especially if you don't remember when the various patches were, so your best bet is to come to this site, find the thread discussing that particular mod, and see if its last update (basically, the last time anyone commented on it) is within the past few months.
Additionally, many mods don't work TOGETHER. Most of the major modpacks are fundamentally incompatible because they attempt to modify the same things in different ways. UI mods are especially tough for this, because instead of incremental additions/modifications (like most game balance mods), these replace the existing game's UI files in their entirety. If you're using a lot of mods, then this is almost guaranteed to occur, although it doesn't always cause an outright crash; it'll just screw up a number of other things in-game in most cases. You can try loading one or two mods, see if that works, and then stack more on until it crashes.
If you're not sure which of the above problems is happening to you, what you should do is turn on logging to see what's wrong. Go into your user directory (in Windows 7, for instance, that's My Documents/My Games/Sid Meier's Civilization V), open up the config.ini file, and look for anything with "Logging" in the name. About two-thirds of the way down the file will be a "LoggingEnabled = 0" line; change that to a 1, and change any other logging ones that sound interesting. I'd also suggest changing the EnableTuner = 0 line at the very top to a 1.
Then, when you start the new game, it'll fill all the log files (in the Logs subdirectory of that same user directory). The ones you care about are xml.log, Database.log, and Lua.log. Each of these will have a few error/warning messages from the unmodded game that you can safely ignore; I'd suggest running one game with no mods, and then save those three files to some side directory just so you have a direct comparison for what the "normal" errors look like. Then, run the game with your mods, and see if there are any additional errors listed in those files. The xml and Database logs will tell you any errors in loading the XML database, while the Lua.log will, as its name implies, tell you if there were any Lua Runtime Errors during the load process.
Also, when you're changing which mods you're using, it's good to delete the contents of your Cache directory (again in that same user directory) to make sure it's not still remembering the mods you've turned off.
It's possible that you'll still have problems even if none of the logfiles shows an issue. For that I'd suggest going to the mod discussion thread for the biggest mod you have, and ask the person there if it's compatible with X, Y, and Z. They'll generally be able to tell you what components of their mod are most likely to conflict with others.
