I need to attach an external microphone to my phone.
Can I do this via the 3.5mm jack on my
Samsung Galaxy S4? What kind of microphone would it be?
You can do that, but you'll need an adapter: the 3.5 mm jack on the phone is for a headset (with earphones and a mic), so it's not made to take a standard mic input. You need to find an adapter that's for plugging a microphone into Samsung phones. It might be tricky to find, but they're inexpensive.
Once you have the adapter, any microphone with a 3.5 mm connector will do. Stage mics requiring XLR and 12 V "phantom power" won't work: you'd need an extra device to supply power to the mic, such as a mixing desk with a 3.5 mm output.