Hey there...to keep it simple:
DAC is a digital to analog converter. Turns ones and zeros into sound.
Amplifiers provide the power to "drive" headphones and speakers, so you can hear your music.
You can't choose "just" one to hear music...you need both a DAC and amp to listen to music. You can, however choose to upgrade one component at a time. A DAC upgrade requires that you have a digital output from the place where you "keep" music...whether it be a computer/digital audio player(ie: ipod)/cd/stereo. An amp requires an analog output (headphone out/line out).
That being said, how are you listening to music right now? Even the most basic computer sound cards or DAP have DACs and amps.
Your choice in how you listen to your music (speakers/headphones) also determines your "needs". Full size headphones have a tendency to need more "juice" than your ipod can provide. In that case, amping really helps. I should back up a second and say amping *almost* always helps. Almost, because the amp may be of poorer quality and introduce noise. An amp may also have a sound signature you don't really like. But in general an amp will help provide a fuller bass, a bigger sound stage, more dynamics and details.
Without more information (including budget), it's hard to say what to get.
If you have a low budget, it may be more wise to save up more money and get better phones, rather than get a cheaper amp/dac.
The truly high end folks say get a separate dac, preamp and amp.
Good luck.