An amplifier-- often referred to as an amp-- is an important piece of equipment in a hi-fi system, whether you're using headphones or speakers. An amp is the second piece of equipment in between your music and your ears, after a Digital to Analog converter (or, DAC). An amp increases the power of an audio signal to a level that your speakers can use to generate sound.
Do you need an amp?
For some headphones and all unpowered speakers, you will need a separate amp. Any device with a 3.5mm headphone jack already has an amp built-in (along with a DAC), but it's not powerful enough to drive some headphones.
If you plug some high end headphones into an under powered amp, they'll sound very quiet. This is where a dedicated amp is needed.
What is the difference between a headphone amp and a speaker amp?
A headphone amp needs less power than a speaker amp and is designed to drive headphones specifically. In general, headphones need less power because they're closer to your ears and much smaller than a speaker setup.
A good headphone amp will have just enough power to drive high end headphones, while at the same time will not introduce any distortion. Many high end headphone amps also offer balanced outputs.
In contrast, a speaker amp has a lot of power to drive big speakers. An non-powered speaker will generally only be connected to the speaker amp, instead of powered speakers that have a power cord as well as the audio cables connected.
Do I need a speaker amp for my powered monitors?
In general, you do not need a speaker amp for your powered monitors. Powered studio monitor speakers take in an audio signal and have an internal amplifier to power the speakers.
With powered studio monitors, you may still be able to use a pre-amp to adjust volume levels or EQs before the audio signal reaches the speaker.