Whether you "need" a subwoofer or not depends on a number of factors, but some of the very best-sounding systems don't use subwoofers.
Back in the days before home theater, only the most devoted bass fanatics used subwoofers. It's not that people didn't like bass, just that they bought speakers large enough to satisfy their bass cravings. Then again, the bass response required to play music isn't as demanding as movie soundtracks, so even fairly small speakers with 4- or 5-inch woofers could produce satisfying bass.
The best of today's small- to medium-size speakers produce much better-sounding bass than speakers did 10 years ago, so if you're primarily interested in listening to music and your speakers have 4-inch or larger woofers, you might not need a sub. If you have a home theater with five or more speakers with 4-inch or larger woofers, they will together produce more bass than a pair of small speakers.
Larger tower speakers, with larger woofers are even better at generating bass, and they don't take up more living space than stand-mounted bookshelf speakers. I recommend using tower speakers for two-channel home theaters and music systems.
I don't have anything against subwoofers, and they certainly can add bass that smaller speakers can't supply on their own. Subs can also help smaller speakers play louder, with lower distortion than they would on their own without a sub. So if high volume and deep bass oomph are what you want, I strongly recommend using a subwoofer in your home theater or music system.
But if you don't need to play loud or feel your room shake, redirecting your subwoofer budget toward better speakers would be a smarter way to go. So if, instead of five $200 satellite speakers and a $500 subwoofer, you were to buy a pair of $750 tower speakers, you'd have much better sound quality. This quality-over-quantity strategy will yield the best sound per dollar spent.