By default, most bittorrent clients will use as much of the available bandwidth as they can get. So, yes, it can definitely slow down your internet. Most of the time you won't notice it for normal web surfing: reading e-mail, reading news, or facebook. But if you're playing games or watching videos online, it will definitely have an impact — possibly a severe impact. If you're sharing a connection with anyone, bittorrent is just plain anti-social.
Fortunately, most bittorrent clients allow you to set up limits. How you do this or what you need to set up depends on the client, and they are rarely limited correctly when you first install them.
Unless you're really serious about your downloads, my personal recommendation is to limit bittorrent to less than 20 percent of your total actual download throughput, and less than 10 percent of your total actual upload throughput. To find those numbers, you'll need to do a series of speed tests to see what your actual values are. Don't just do the test once: re-test multiple times at multiple times a day over a period of a couple weeks. Use the lowest numbers recorded and limit your bittorrent to 10% of that.
Of course, if you're more serious about your downloads this won't work for you. But if you view it as more of a background process, this works well.