While explaining a major triad as scale degrees 1, 3, and 5 is accurate and simple, it is lacking. Major triads are not strictly limited to this combination. For example, degrees 4, 6, and 1 also form a major triad, as well as 5, 7, and 2, and if we want to get fancy, we can even dare to steal from other keys and use b2, 4, and b6. Therefore a more comprehensive definition is one that includes the intervals within the triad that construct it, because these intervals do not rely on the context of a key.
First things first:
A triad is a set of three notes in a neat little stack of thirds. That is, the distance between the bottom note and the middle note is a third, and the distance between the middle note and the top note is a third. In turn this means that the distance from the bottom note to the top note will always be a fifth. On a staff, triads spelled as chords (all three notes played at the same time) will look like snowmen: three circles on top of each other, the same distance apart (all on lines or all on spaces). It should be noted that triads do not have to be chords. They can be written (or played) linearly (as in a melody), or even out of order, and still be considered triads.
Triads come in different qualities: major, minor, augmented, and diminished. What determines the quality of the triad is the quality of the thirds within it. If there are two thirds, and each can be major or minor, then you can see how there are four possible triads to be made. One would consist of two major thirds: this is augmented. One would consist of two minor thirds - diminished. But you can also have one of each. In these cases, the triad is named after the quality of the lower third; therefore, if the bottom third is minor, and the top third is major, that creates a minor triad. And finally, if the bottom third is major, and the top third is minor, we have our major triad.