Okay, firstly downloading copyrighted files from limewire may be illegal in your country (unless you happen to be French or Canadian). So you can get sued by the RIAA or MPAA or whatever anti-pirate group that chooses to sue you.
Furthermore, limewire (frostwire, carbos, kiwialpha) operate on the gnutella network on which there are a small number of fake files and even pornography so you have to discern if the file is legitimate or not. You can always check by right clicking on the file -> advanced -> bitzi lookup. Plus if you disabled your firewall or router to use the program then you may be at risk. And do check what files you are sharing to ensure that you are not giving the world access to private or confidential files.
So when you use limewire keep track of three things: Is it legal, is it right? And is the file I am downloading the real thing? And lastly, am I opening any security holes? If you keep these in mind, then downloading from limewire is safe so long as you download legal and legitimate stuff.
Now on your question on what bittorrent is. Bittorrent is the name of a p2p protocol which uses swarm technology to facilitate bandwidth efficient downloads. For example, when more users download, the more upload and hence the torrent is healthier. The name Bittorrent also refers to the official client which uses the protocol. Both the official client and protocol were developed by Bram Cohem. You can visit the official website here: http://www.bittorrent.com.
However there are other clients out there such as azureus, utorrent, bitcomet, bitlord, bittornardo... But I would recommend you to download utorrent as it is small and efficient. Unlike other protocols and the clients that use them, Bittorrent's protocol does not give the option for searching the entire network.
Bittorrent works in this way, the distributer selects the files he wants to share and creates a .torrent file to give to other people to tell them what his file is and where to download it from. He can use a tracker or dht as the tracker to regulate the download, connect nodes to one another, etc. The place where the file is downloaded from is the distributor's computer which is the first seed (the name of the guy who has a complete copy of the files and is uploading them). Other people than open the torrent file with utorrent, etc and connect to the uploader, they then request the file. (These nodes are referred to as leechers). At the same time these leechers are also uploading the file. Once they finish downloading the file, they are called seeders.
Bittorrent uses a tit for tat download system, the faster you upload the faster you download.
I hope this helps!