If you're looking at the sheet music, it's the key signature. To determine if it's in minor look for an unusual amount of naturals and sharps. For example, no sharps or flats could be C Major or A minor. If you see a lot of G#s and you end on a lot of A, C, or Es, then you're probably dealing with A minor rather than C Major. The sound gives it away too. A more happy consonant sound leans more towards major keys and a sad dissonant sound leans more towards minor keys.
If you're doing it aurally, it's a matter of looking for clues to figure out what key it's in. The few clues I try to listen for are the cadences, leading tones, and runs.
The runs are usually the easiest to determine, as long as it isn't chromatic it'll work. In an ascending or descending truncated scale you'll probably hit some crucial notes to help with determining the key. Figure out the first or last note of the run and count how many notes were in the run. Then count the notes forwards or backwards, and figure out if the notes you're playing work with the run you're hearing. Then you'll have a section of the scale and you'll most likely be able to figure out the key due to the sharps or flats, or the lack thereof, you found in the run.
Leading tones are the 7 leading to the 8/1 (or the 7th note the scale leading to the 8th note of the scale). Which are sometimes easy to identify because it's usually associated with a chord that is creating a tension that wants to be resolves V7 to I, etc. Though, it could be a secondary dominant or a secondary leading tone. Or you may pick out the 4 leading to the 5. But, through trial and error you'll find a scale built off of those won't fit the rest of the song.
Cadences are the chords right at the end of a phrase, the last two generally. If you get that finite feeling, at the end it's probably a I chord. Which, coincidentally, has the same notes as an arpeggio. Once you figure out the notes of that I chord (listening for the lowest note is usually your best bet for your 1) you've got the bare boned scale, so you're set. The I is also like the reset button after certain crazy phrases, so if it feels like a reset, it's probably the I chord. Common cadences are V(7) to I, IV to I, IV to V(7), vi to V. If it feels like the phrase wants to keep going, it's probably a V. If it tricked you into thinking it was going to end, then you probably have a V(7) going to a vi, otherwise known as a deceptive cadence.
Being able to identify chords and cadences will make figuring out keys and stuff, tons easier. If you ever get a chance play some chords and do some chord analysis on pieces. Music theory can be difficult, but like math you have to start from the fundamentals and work your way up, otherwise it'll feel like learning rocket science when you only have a vague understanding of basic algebra.
Having a piano handy for this makes life a million times easier, btw. Don't be afraid to just play notes until you figure it out. Eventually it'll click.
Songs do modulate and change key throughout sometimes. It's easiest to determine the key it's modulating to at the modulating point. So if you can find that point and catch that weird chord leading to the I chord of the new key, you'll be set. There are also modes and other kinds of scales, but those are something you actually need to read up on and learn about because they have their own patterns that aren't as easy to identify as the major or minor scale patterns.
This became a wall of text, sorry about that, but hopefully it answers your question?