Malware refers to malicious software, and software that does something with a malicious intent, with anti-malware being about taking a stance against it. Malware comes in a variety of guises; there's viruses which spread by themselves between systems causing problems, trojans which hide inside another program that you use without knowing about the hidden functions, malicious software that does something under someone else's direct control, and the list goes on…
As I've already said: In general, any system that I've seen where the owner doesn't bother with any anti-virus and anti-trojan effort has been infected several times over. Not only does it make their system unstable, or unusable, it affects other people that they interact with, and complete strangers.
Doing something about it is an on-going battle, and it's something that you have to do for yourself. You need to be cautious about what you put onto your computer, and you need to check what you do put there. It is possible to find free software to do this, but because of the research necessary into dealing with malicious software you can expect the better protective programs to cost you something.
This is software that detects known viral programs, and may also be able to recognise patterns familiar to such programs, but not currently identified specifically as a virus. Usually they can also repair the damage caused, to some degree.
Often anti-virus software will only deal with viruses, you'll need something else to deal with other types of malicious software, but I've found My e-Trust's anti-virus software does also detect trojans (in my case, it's been finding them arriving in spam mail), and it isn't a major drain on the computer, as well as being fairly reasonably priced.
Software that detects trojan software (malicious programs hidden within something else). While I've not tried it, I've heard that “TDS” (Trojan Defense Suite) is very good.
I haven't been compromised by trojan software to have a need to use anti-trojan software, because I'm careful with what I do with my computer, and I use software which doesn't compromise my computer for me (i.e. I avoid Internet Explorer and don't use Outlook Express, I use much better alternatives). Yes, I've received trojans in the mail, but they haven't been given a chance to do their business, they arrive as ignored data.
Spyware is something in a program that keeps tabs on what you do, and reports it back to someone else (hence the name). This could be done by something malicious that snuck in, or an unavoidable extra feature of some software that you want to use. You use anti-spyware programs to detect and remove such things. I've found Lavasoft's “AdAware” program to be quite good at this.
In some cases you can remove such extra programs without killing the program that you want to use, but not always. In those cases, an alternative approach can be to simply firewall off the communications made by the spyware.
While the spyware may not be malicious in itself, the intent behind it might not be so benign, so I do lump spyware in with all the malware. And it's not uncommon for such programs to be a heavy burden on your computer.
It's possible to spy on you without using software on your computer. For instance, some websites include content from yet another website which exchanges cookies with various other interelated websites. Once they've done that, they can build up a profile on your browsing habits. Some anti-spyware programs will detect and remove such cookies for you, but the damage has already been done by that stage.
Adware is advertising supported software. And while advertising may not be malicious (though it is possible), you don't know what they'll do with the data that they collate on you (if they do that), and you have to be careful that software with advertising is just software with advertising, and not with something else as well. It's also not uncommon for adware to be snuck in with something else, and for there to be no way to remove it yourself.
Because of those concerns, I do lump it (adware) in with the other malicious software. Again, Lavasoft's AdAware is a handy tool for dealing with this sort of thing. Indeed, my only real need to use their program was to deal with some advertising crapware that was surrepticiously included with something else, and not removed when I removed the thing it came with. AdAware is fairly good at differentiating between adware that you would have agreed to install compared against surrepticious installations, or installations that were devious about their intentions.
I've mentioned some specific software, but there are plenty of alternatives, some much better than each other, and it may well be that you need more than one to cover all the possibilities.