My current firm is loathe to let go of me. When I went to the partner in charge of my practice group to ask for a reference for my job search, he offered to match the salary that any firm offers me. How should I handle this?
In terms of your long term career, there are many advantages of sticking it out at your old job, especially if you are going to be equally compensated. First, if they like you enough to make a matching offer, then they will probably treat you pretty well if you decide to stay, maybe even let you sneak out of the office before 8 pm on a Friday night. If you leave your firm, you will be starting over and you will have to prove yourself yet again, and since you have already made a great impression at your current firm, maybe that isn't something you want to have to do again. Second, it sounds like you have an excellent chance of making partner at this firm if they are that interested in keeping you. In case you are under the impression that your firm does this with all of their associates who try to leave, let me clear that up without knowing anything about your firm - they definitely don't. This is extremely unorthodox. Third, if you eventually do want to leave this firm, the more time you spend in one location will make your resume stronger than if you continually are moving around.
The advantages of moving to a new firm depend entirely on the firm you are considering. A more prestigious firm is always a lure with attorneys and can be a good career move for the long term, as well. A change of scenery and a change of the type of work you are doing are some other reasons why moving might be a good idea. One of the main reasons that another firm's offer could be more enticing than your current firm's is that at a new firm, you can remold your image. As previously discussed, having to reprove yourself can be awfully tedious, but if what you really seek is to make it so they don't expect the same things of you (e.g. your willingness to handle grunt work without complaining, your willingness to work national holidays, etc.), then moving firms may be your only option.
Either way, you clearly have a lot to think about. The most important consideration, though, is whether or not more money is what you were after in the first place. If not, then you should most likely not consider your current firm's offer at all.