I'm using Bitcomet and love it but Azureus and uTorrent are two great bittorrent clients as well, and I know that a lot of people are using them. The AureusWiki site has an entry that tries to help you download your torrents faster. They post some interesting links in that article that deal with common problems and settings that slow down downloads. They provide a chart for example that take different upload speeds and define other settings that are optimal for good download speeds.
JohnTp published an article that tries to explain how to increase download speeds with uTorrent, another popular client. This article is really helpful, besides a step by step walk through, John added screenshots as well to visualize everything even more.
Just a side note, you will never be able to download at greater speed than your connection is capable of, that means: If your upload or download speeds are already near the maximum speeds of your Internet connection, it is not necessary to tweak the settings any more as you won't be able to benefit from the changes that you make. It´s great for everyone else who is having troubles.
Update: John's website has been taken off the web, and we have removed the link pointing to it as a consequence. You can try the uTorrent Connection Setup guide as an alternative, as it offers some helpful tips on how to configure the bittorrent client. Especially the problem section may come in handy, as it explains port forwarding, how to resolve NAT problems, or how to use universal plug and play, or nat-pmp.
Here is a quick rundown:
Ghacks is a technology news blog that was founded in 2005 by Martin Brinkmann. It has since then become one of the most popular tech news sites on the Internet with five authors and regular contributions from freelance writers.