Well, there's actually two mechanisms to download a torrent - one is a file, containing information on trackers, and the other is a "magnet link" which is a link which the torrent client uses to work out where other clients are. For our purposes, these are pretty similar.
Your torrent client is a bit of software that collects bits of the file you want to download from other torrent clients - I have an explanation of the process here.
Ideally you will want to set your torrent client as the default handler for your torrent files or magnet links - the installer will do this or you will have to open the torrent file manually - The process would depend on your OS - but treat it as you would any other file with a known program to open it with. Rightclicking and open as would work in windows, or opening the torrent file from the file menu of your torrent client, or picking the torrent client as the way to handle links of the type or files of the type from your web browser. When you open the file, it asks you where to save it, and starts downloading.
As for different clients, bittorrent is the 'official' client, based off utorrent these days (with utorrent being the client of choice for quite a while. Sadly, Its moved away from its old standards of being tiny and simple, and is ad supported these days). I migrated to qbitorrent since utorrent had ads. That said, any modern, standards compliant torrent client would work the same way.