Summary:
I have been at my current job since mid-February 2013 and I have a job offer from a different company. Should I tell my current company about this offer before I make a decision? I would be willing to stay for salary increase.
Details:
I interviewed with company A (for Senior Accountant) and B (for Associate Senior Accountant) in February 2013. A did not offer me the job and B did; I took job B. In early May 2013, recruiter at A asked if I wanted to come back in to interview for Staff Accountant. After 4 interviews spanning about 5 weeks, I have been offered the job at A. Should I inform my current company (B) that I have been offered a position with A before making the final decision? A salary increase would change my mind.
Notes:
Both companies are technology start-ups, A being much bigger.
I am not concerned with title (Senior at B to Staff at A), I am concerned with salary

No, don't do it
Make your decision of which company you want to work for and then work there. If I was your manager and you were interviewing and had a job offer and you want it, then take it. I never hold anyone back but if you come back to me and ask for more money, even if we give it to you, I don't trust you will stay and will be figuring out a way to replace you as I will assume you are flakey and I need employees who are committed. If it's all about the money then go to the other job.
Of course, there is a podcast for this. Just cannot recall which one. Perhaps someone else can help.
Never, *ever* use an offer as a salary negotiation tactic
If you want more money at your current job, then tell them that. You don't have to add "or I'll quit", because that's implied. When making your request, you'll want to provide information on *why* you're *worth* more money. Companies don't pay what you want/need, they pay what makes economic sense to them in order to get a job done.
As dtiller said, if you go to your manager with an offer and say "I'll stay if you match/beat this", you're forever marked. You're going to be the last person considered for promotions, special assignments, and anything else that is either a perk or requires someone committed to the company. You're also going to be the first person on the block if they need to downsize or "restructure". If you're really unlucky, and you've got an absolute nightmare of a boss, you may even be fired, and then be stuck with an incredibly unfavourable reference.
As far as casts go, I think the recommendation not to solicit a counteroffer has been discussed in a number of casts, but I don't recall (nor can I find) a cast dedicated to the topic. The closest I can find is http://www.manager-tools.com/2011/11/what-do-with-a-counter-offer which from memory deals more with unsolicited counteroffers when you've handed in your resignation, although I'd be surprised if it didn't mention why soliciting is a bad idea in there somewhere.
interesting points
Matt made an interesting point about being marked forever. Just pick the company you really want to work for. Do your research. In the past year I have learned that some companies have a bizarre cult like culture and may be upset for you leaving so early, but you have to do what is best for you.
Thank You
I'm so impressed every time I write a post here, thank you all so much. I've decided to go with the new company because I think I'm a better cultural fit, i am getting more salary/equity, and it has similar if not better growth ahead. I've also had a bad review at my current company and I don't feel like I can dig myself out of it very easily, if at all, which has also had an impact on my decision because I don't feel valuable. A follow up question I have is this: I'd rather not say who my new employer is to my old employer, but I don't want to come of as snide, how do I go about this when my manager, HR, and friends/employees ask me this question?