I bought a bottle of balsamic vinegar, and it smells so sour when I put my nose over the bottle.
Is it supposed to be like that?
Well, it IS vinegar. However as you probably already know, higher quality balsamic vinegar is less sour and more rich tasting than its cheaper counterpart. It's unlikely that your balsamic is spoiled--in most ways vinegar is as spoiled as it's going to get.
More likely is that you've uncovered one of the issues with balsamic vinegar: Most of what you can get in a supermarket is of low quality, and is not produced in the traditional way (years of curing and evaporation in wooden casks). What you can get is more likely to be much more mundane wine vinegar with flavorings. And that is often more acidic-tasting and pungent than the real thing (which costs an arm and a leg).
I'd say sample it carefully and see if it's as sour as it smells. It might be OK, or you might want to find a new brand or get it at a store that goes through it faster (check the best by date, if there is one).
I have found what improves a cheaper balsamic vinegar is to remove the top, rubber-band some cheesecloth on the top, and put it in the back of your pantry for a year or so.
There's a good chance your vinegar went bad, though vinegar of all qualities can usually be used while cooking. Expensive vinegars can be used straight, such as with cheeses, as a sauce over pasta, or glazed on fruit.
Lower quality vinegars are better for cooking (Put balsamic in towards the end of the cooking/heating cycle, as the flavor will dissolve when heated too much!) and mixing in foods.
How do you know when a balsamic is good quality? Yes, there are stamps of approval from balsamic boards in Italy, and high-quality balsamic vinegar should only come from a few of these regions. When you pay more for a good balsamic (price is a good indicator), you'll often get a wax seal, a sweeter flavor from aging, and a much more viscous vinegar. The older, more expensive, and more aged it is, the more like honey it will look when you pour it out of the bottle. Runnier, looser balsamics are diluted with water and will often taste more tangy (as does yours--though again, there's a good chance it went bad). Try Dodi brand or balsamics from the Emilio region, especially those that come with certificates.
Great balsamics are aged, thick, sweet, and sticky, so although yours is probably fine for cooking, I'd invest in a higher-quality vinegar for special occasions.