What type of Trojan you may encounter depends on what "activates" it. If you go back far enough (think MS-DOS on old PCs), not only did you have to download the Trojan, you had to choose to run it. At one time, someone had released a Trojan of the very popular PKZIP app. It didn't zip or unzip anything. The person who released it to bulletin boards, did so as a version number that hadn't been released as a real version yet, so everyone thought it was a new, legitimate version. I can't remember what it did when you ran the app.
Currently, most known Trojans are obtained in these two most common manners:
1) Illegal software downloads. These contain the nasty ones like keyloggers, back doors, etc.
2) Legal software from sites like Softonic, C|Net's download.com and others. These aggregate sites where you can pick up all kinds of software to try have been adding extra installers to just about everything you download from them. Essentially, almost 100% of these unwanted additions are adware. Harmless, but greatly annoying. Especially the more aggressive ones that put so many adds on the screen, and so heavily tie up your Internet connection retrieving ads that your entire system slows down.
Either of these when installed are immediately active.