The regular download window, referred to as "direct download" is simply you downloading a file stored on the server that is currently hosting the website you're browsing.
Torrenting involves downloading a file, (.torrent) that is usually pretty small (less than 1 MB). This file contains the location of the bigger file that you want to download, e.g. a movie, or game. That movie or game is stored on other people's computers - and your computer uses the torrent file (and a BitTorrent Program) to connect directly to those computers and download the bigger file.
It is not possible to download torrents through a regular download window, simply because you cannot connect to other computers that host torrents, using your internet browser.
As for your question about magnet links:
Azureus is a BitTorrent client, which is a program that lets you download torrents using a torrent file (that same .torrent file I mentioned earlier). With a magnet link it becomes possible to download the .torrent file from another user's computer through Azureus' DHT network. The main purpose of this is to give people an alternative way of getting the torrent file should the website hosting the torrents go down.
It seems to me that you're looking for a BitTorrent client to download torrents, one that is fast and easy to use.
I recommend uTorrent, which can be downloaded here: http://www.utorrent.com/download.php
Two things you can do to improve your download speed are: Port Forwarding, and downloading PeerGuardian 2 (if you're running MS XP or lower).
Port Forwarding greatly improves your download speed by allowing more people to connect to your computer. Learn more at http://www.portforward.com
PeerGuardian2 is something that I would be greatly ashamed of myself if I didn't mention, as it protects your computer (somewhat) from being pinged (or found) by government agencies, as well as prevents you from downloading bad data from malicious peers. You can download it here: http://phoenixlabs.org/pg2/
Need anything else, send me an email.