This post targets students at the London School of Economics specifically, but may be of use to other London- or UK-based students completing degrees in the humanities. The websites listed are primarily focused on work (including internships) in the UK, though not exclusively so. There are dozens of websites out there--these are simply my favourites (and those that I know my students have relied on over the years).
Best for politics
www.w4mp.org is great for jobs in politics--not only those in Parliament (though this is probably the best place to go to look for full- or part-time work at Westminster), but also for jobs in, for example, think tanks or other politically-oriented organisations or political parties. (As an aside, I've heard it's also a good place to look for flatmates!)
Best for the non-profit/ NGO sectors
Many big-name development agencies, non-profits, NGOs and charities advertise jobs through www.idealist.org. A plus is that you can sign up to their database listing your interests and get email updates whenever relevant jobs get listed.
Best for journalism
This isn't my field, but my students 'in the know' on this highly recommend the "jobs" section of www.journalism.co.uk.
Best for academia/ research
www.jobs.ac.uk also has a useful system whereby, once registered, jobs in your area of interest are emailed to you. You can also upload your CV so employers can find you (though truthfully I've personally never heard of anyone having success with such a tactic).
Best for variety and scope
The Guardian newspaper of course has a reputation for certain political leanings--however its job website http://jobs.guardian.co.uk/ attracts employers from a huge range of sectors to advertise jobs that are going--so regardless of what you're looking for, I'd recommend checking them out. They also have a section for volunteer positions.
Best for finance
www.cityjobs.com has a good reputation for those interested in banking, accounting and finance.
Best for temp work
Many students don't realise that there are temp agencies out there that focus exclusively on certain sectors--such as the non-profit sector. There are many reasons you may find yourself needing to temp, but it needn't be solely a money-making endeavor: you may find yourself doing basic admin work such as answering phones or filing, but it can still be a useful to get insider experience of an industry, get your 'foot in the door', etcetera.
As two examples of temp-specialist agencies that I know my students in the past have worked for:
-Charity People places temps within the non-profit and NGO sector, with clients such as UNICEF and the British Heart Foundation.
-For those interested in higher education or research, Prospect-Us has a good reputation. (For example, the LSE sometimes uses temps from this agency).
Best for internships
I promise this is not a shameless plug for my own employer, but I hear that theLSE's Career website is often the best place to find internships (and also jobs). Many excellent employers come to LSE Careers in order to advertise their available positions. By going directly to the LSE, employers know that candidates will likely be 1. high quality and 2. interested and educated in the relevant required subject-area. This service is available to current students and also alumni, and it is worth registering your details with them. (For non-LSE students--of course, this applies to your own Careers Office--your university has a vested interest in getting their students into good positions, and will usually be very helpful and informed where your particular interests and skills are concerned.)
With regards to internships, I would also just say that many of my students have landed them by going directly to an employer they are interested in--either by looking at their websites (most will have a job/ internship section) or, in the case of smaller organizations, contacting them 'cold' with your CV and expression of interest. Particularly where unpaid internships are concerned, sometimes organizations/ MPs/ charities etcetera don't advertise posts but may create one if you go to them with enthusiasm and a clear time-commitment and interest.
Finally, a note to those who are still studying and who will be finishing in say the next 6-9 months: I want to emphasize that many of you will not yet be thinking about your next career move, and I think that is completely fine. I did my MSc at the LSE in 2001, and didn't know until summer 2002 what I wanted to do next. There is a lot to be said for enjoying your time studying, following your interests, and just 'being a student'. As such, this post for those who are already actively looking for jobs, but also highlights websites that you can sign up to receive job alerts from throughout the year--not necessarily so that you can actively apply for them, but because I think it's useful to know what jobs are 'out there' before you finish your degree.
Happy hunting! And please do feel free to share below any others I have missed off--and perhaps any others that are good for jobs outside of the UK.