What's the difference between a DAC and an Amplifier?
A Digital to Analogue Converter takes the 1's and 0's of the digital music and turns into an analogue waveform, in other words, the sound you can hear when played through a transducer, as notes or whatever. An amplifier then amplifies that analog waveform into a signal that can drive a transducer or speaker. A line out signal straight out of a DAC is mostly voltage, so assuming you can wire that up, it will be too loud, and dynamics aren't reproduced well enough because it has no provision for the current demands of a headphone or speaker. An amp does more than control the sound - that's the job of the preamp stage on the headphone amp - but like I said it amplifies the signal into one that can run a speaker or headphone.
Now a speaker or headphone will have current demands as well as voltage because of the physics that leads to it reproducing the analogue waveform into actual sound. For starters, its efficiency rating may require a certain amount of power to hit a volume level that a listener needs or prefers. At the same time, the nominal impedance that a headphone or speaker is rated at is just that, nominal - meaning it isn't always fixed at that imepedance level. If the transducer plays a bass note for example the impedance might drop, and ideally an amp should produce double that power at half the impedance to produce the proper transients. This ratio is more easily achieved in speaker amps because the range of speaker impedance is very narrow - mostly 6ohm to 8ohm, then some 4ohm, maybe 12ohm and 16ohm. With headphones it gets more complicated as they can range from 8ohm all the way up to 600ohm. This is why you might encounter for example some amps as "current-driven" designs, which is basically what speaker amps tend to be,* as opposed to ther amps that are voltage-driven. Now, it's not so much you only have just one, but it depends on how much more of the other. Typically low impedance means more current is necessary, but if your headphone is relatively less efficient, then it will require a lot of voltage still; high impedance typically means it needs more voltage, but if it's efficient enough, then it won't need as much voltage. Such an amp might make more power at 300ohm than at 150ohm and 32ohm, but that rule about impedance and current doesn't apply in the same sense here.
*Marketing n speaker amps say "High Current" or a fancier version like "WRAT - Wide Range Amplifier Technology" which in some cases might not actually be a patented circuit design, but just a fancy way of saying "high current"