All the job postings from recruitment agencies I pop into don't mention the name of the company the agency is recruiting for. I was wondering what is the reason for this? I think that everyone will want to know the name/details about the company before applying, so everyone will need to contact one of the agency's recruiters to get that information anyway. Is it because they are afraid that another agency will steal their clients? Or they think that people will apply to the respective company without their help?
Both, is the short answer.
Some agencies/recruiters will try and steal clients from others, or it might well be that multiple recruiters are hiring for the same role.
In addition, it costs a company less to hire you directly (they're not paying the finders fee, and salary cut to the agency for starters).
There's no benefit for agencies to reveal the company until you're expressing an interest.
Something else that happens (apart from the poaching in ALPHA133's answer) is that quite often a (less reputable) recruiter will hear about a client's open role, and will gather resume's to bundle up and (hopefully) get a slice themselves. In this case the client will usually be carpet bombed with a load of redacted resumes in the hope that the hiring manager will agree to some high fee to retain the recruiter who has so much "talent".
In this case the recruiter won't be able to tell you who the client is, until they've got the contract, at which point they will approach you as if they've just suddenly thought of you for the role (when in truth the client has already seen some form of your resume without your knowledge).
I used to get this every time I advertised a role in the UK, from one of the largest recruitment agents who were notorious for it, wanting "only 25%" if I hired someone (the agencies I worked with charged around 10%).