Note: This is a description of my experience. I do not guarantee this will work for you. Please check with the UK Border Agency before applying. If you have further questions about the process, please comment on this post and ask.
Target Country: UK
Purpose: Work (Post Study)
Processing Office: Cardiff, UK
1. Filled out form (See UK Border Agency Website). It is important that the applicant has enough points - this is a point based system.
2. Letter from University stating that I have successfully completed the course (It has to be an approved course from an approved institution). This letter can only be issued once you have graduated. My college had the option of graduating early -- those who graduated early, were able to apply for their visas early.
3. Bank statements from last 3 months - showing that I had at least £800 in my account for the last 3 months before applying. A printout with the bank's stamp on it sufficed. (Note: If applying from outside the UK, you have to show £2800 for a single applicant for 3 months. The amount is higher if you have dependants).
4. Two passport sized photographs, as per the specifications of the Border Agency. Any regular photographer in London should be able to give you these.
5. Current passport
6. My older passport (this had my student visa, but the passport had expired).
7. £800 processing fee (I made a premium application in person for fast processing. Normal postal applications within the UK cost £550 at present).
Process & Experience:
It is possible to get the Tier-1 visa in 1 day. For this you need a premium appointment at one of the UK Border Agency's offices. These appointments are very difficult to get. You must book one a month in advance. It also helps to ring everyday and hope someone cancels (this is what I did!).
After ringing for 3 days in a row (ringing a premium number, set me back by £20 in total), I got an appointment at the Cardiff office. The catch was that Cardiff cannot print visa labels. While they can approve/reject your application on the same day, they normally send your passport by courier to Sheffield (or another office) for pasting the visa in your passport. I got my passport back 3 days after I applied.
The people on the phone were very friendly and helpful. I have never spoken with more helpful, efficient and polite government employees. After confirming my appointment, an officer from the Cardiff office rang me, and made sure I understood all requirements. He was extremely polite too.
Upon reaching the Cardiff office, I was let in by the security guards (who held on to my phone and other electronics). I recommend carrying a good book, since you may have to sit around for some time. After a 5 min wait, the officer called me, checked my application and asked me to pay. After paying, I went out and sat in a cafe for 1.5 hours. Then I had to return to the office and wait for another 30 mins. Then the lady came and gave me a letter saying that my visa had been approved. And I was free to go.
The passport came 3 days later by courier.
Overall - a very pleasant experience. The people at the Cardiff office were very helpful and friendly. A little more expensive than the normal postal route, but you save almost 6 weeks! So totally worth it.