You should be able to get a reasonable steak stovetop using a cast iron grill pan, if you have a strong enough exhaust. Oil the cast iron pan (with canola or such), then heat it very hot, until it starts to smoke. Make sure the meat is completely dry on the outside (wipe with a paper towel, water will prevent browning) and gently place in the pan. Leave it there for a minute or two (it'll smoke quite a bit!), rotate 90° to get the nice grill marks. Leave for another minute or two. Flip, and repeat for the other side.
It may splatter, have a splatter screen handy. It will smoke, quite a bit, make sure the exhaust is on high.
You will probably have a medium-rare steak now. Using a thinner cut will make it more well done (you can cut a thick steak in half with your chef's knife, making two thin steaks); so will plopping it in the oven (not sure if you want the oven before or after searing, I like 'em medium rare...).
Remember to let the meat rest for 5 minutes or so before serving.
Also, if your steaks are coming out ridiculously tough, you're probably using the wrong cut of meat, or some terrible grade. What cut are you using?
[edit: I should note that, in case it wasn't obvious, the burner should be up pretty high]
Edit by rumtscho There is a reason this answer specifies a cast iron pan. If you are limited in your cookware choice, pay attention to the maximum temperature your pan can tolerate. If you are using a non-stick pan, you have to go on medium heat and wait longer, else you'll damage the pan. (And sorry, but you can't get it as tasty as on high heat that way).
The only real bacterial problem you have to worry about with steak is e. coli, which lives on the outside. Since even in a blue application the outside is cooked sufficiently to kill any bacteria, you should be fine. (Presuming you are using fresh steak, naturally).
For blue: get a pan screaming--and I mean screaming--hot. Toss the steak in to sear, flip to sear the other side. Do edges if needed. Serve immediately.
For rare, just do the steak in a normal pan or grill, flip once, about two minutes per side (less if the steak is very thin).
For both blue and rare applications you will want steaks that are on the thick side, and with not much marbling (intramuscular fat), as the short cooking time won't melt the fats. I suggest tenderloin/filet.
Best Answer
In this thread, Lorenzo notes that steaks may end up raw in the center if you only leave 'em out for 10-15 minutes to come up to temperature. I recommend 30 mins, but it will probably vary on what room temperature actually is for you.
From Weber's Way To Grill, which recommends salting right when you take them out of the fridge, here's Mr. Purviance's take on why you should consider salting early: