I thought I'd dedicate a whole post to this because it's something I had a lot of questions about when I was researching how it all works. So hopefully I can be of some help if you're looking to do the same thing. The following information is correct as of December 2014.
There are a plethora of websites offering Vietnam visas through online applications, and then you get stamped in on arrival at an international airport. However, the official Vietnam Embassy website says that these aren't legit and you need to apply in person at an embassy. I've heard stories of people doing it online and being allowed into the country, but I didn't want to risk it. The same with doing it through a travel agent on Khao San Road - it's safer and cheaper to do it yourself.
When I got to the embassy in Bangkok, I did see a sign that said that you can now apply at the official website. This is obviously very good news, and makes the process a lot easier.
However, if you still apply in person at the embassy like I did then this is how it went:
The Vietnam Embassy in Bangkok is on Wireless Road, near Lumphini Park (map in Thai here). The closest BTS (overground train) is Phloen Chit, or Lumphini station on the MRT (metro) is a bit further but still walkable distance.
The opening hours seem to vary depending on how they feel (as with much in Southeast Asia), but 8.30am-11.30am, and then 1.30pm-4.30pm seems to be about right.
I arrived at 9.45am and filled in one of the paper forms they provide. They don't have pens though, so BYO. They also provide glue for you to attach one passport photo to your application. If you don't have one, there's a photo shop near Phloen Chit BTS station that is very well equipped to do them for you. Just ask for a Vietnam visa photo and they know what size is needed. It costs 150 baht for four photos, although I only needed one.
The shop is called Closeup Photoshop, and when you exit Phloen Chit station take Exit 2, make a U-turn at the foot of the stairs, and it's on the right near Mahatun Plaza. You need to turn onto a small road to the right off the main road and then it's on your right hand side. Literally less than a minute's walk from the BTS station.
At the embassy, I asked for a same-day visa, and the man told me that it was too late in the day to do same-day (this was at 10am, and I'd read that they do them until 10.30am). I kind of got the feeling that they can do it at that time, he just wasn't going to cooperate.
Thankfully next-day collection seemed fine. Maybe ask for same-day and then they'll offer you next-day as a consolation prize. The bonus was that my collection time was 11.30am, so I still had time to catch a bus to Kanchanaburi in the afternoon afterwards.
It cost me 3000 baht for next-day. It's cheaper for 4-day collection, but I think it's still over 2000 baht. Not sure what they charge for same-day, if you're early/ lucky enough to get it.
I came back the following day at 11am, and was able to collect my visa late at about 11.50am. But it was all fine - I checked all the details of the dates and my passport number etc. So it's happening; I'm allowed into Vietnam!
Incidentally, you can also apply for the Vietnam visa in Phnom Penh or Vientiane. But I think Bangkok is just a popular place to do it, and they seem very well set up to process them relatively quickly.
If you have any questions, comment on this post and I'll try my best to help out!