Answer: The major intervals are the larger ones - the words basically just mean "large" and "small."

There are 4 interval types that can be either major or minor:

3rds (major and minor thirds, like C upward to E or D upward to F)
6ths (upside-down thirds, like E upward to C or F upward to D).
2nds (like C up to D or E up to F)
7ths (upside-down seconds, like D up to C or F up to E)

There's an emotional implication too, especially in the case of thirds and sixths. The major thirds and sixths are traditionally regarded as being positive or "happy" while the minor thirds and sixths are associated with darker moods, perhaps because the ear can tell the minor intervals are just a tad short of being what they could be: the major intervals are acoustically stronger. This acoustic characteristic is the most pronounced in the thirds, then the sixths and seconds. The minor and major 7ths don't have quite the same associations.

So, how can you tell? If you're telling by looking, you'll begin by being aware of the size of each interval as measured in half steps (the smallest step on the piano). A minor third like D-F covers three half steps. A major one like C-E covers four half steps. Minor sixths are a fourth plus a minor third, major sixths are a fourth plus a major third; a minor second is the smallest step you can make on a piano, while a major second covers two such steps.

And after a while you just know those without counting.

(The terms "second" and "third" refer to the number of letter names included in the interval. A second is two names, C-D, or E-F... a third covers three: C (D) E, or D (E) F... but C-D is bigger than E-F and C-E is bigger than D-F).

To tell by ear, you begin by learning the difference in sound between a major triad and a minor one. The major triad (such as C-E-G) begins with a major third; after a bit it will be plain to you whenever you hear a major third that this could be the start of a major triad. Same for a minor third as part of a minor triad (like D-F-A). It also helps to step to the interval as part of a scale or tune: Are you Sleeping, Brother John - if you remember that - outlines a major third: C, D, E, C, C, D, E, C... The old movie and even older TV Series theme "Dragnet" is very similar for the minor third: C, D, Eb, C...
See Question 45 for more on that topic.