1,3,5 refer to the notes that make up the basic structure of a chord known as "the major triad" which is derived from "the Major Scale." (It's called the major triad because that particular chord is only made up of 3 tones) The major triad is the building block for all major chords (or for all chords for that matter, but for simplicity's sake let's just deal with major chords). Use the C major scale as an example:

C, D, E, F, G, A, B, C
1 , 2, 3, 4, 5 , 6, 7, 1

The numbers 1,3,5 would refer to the tones C, E, and G in the above example. The C major triad would be comprised of the notes C, E, and G. C would be the 1st tone (or the root note) of our chord. The E would be the 3rd tone. the G would be the 5th tone. You may have come across chords such as, C7, C9, C11, C13, CMaj7, C6, etc... in the course of trying to play a piece of music. For now, just realize that all of the chords mentioned above (and many others) are built on the major triad. This means that all of those chords contain the 1st, 3rd, and 5th tones of the major scale. Using othe above example, all of the chords I mentioned contain C, E, and G.

C7 -- C, E, G, Bb

C9 -- C, E, G, Bb, D

C11 -- C, E, G, Bb, D, F

C13 -- C, E, G, Bb, D, F, A

CMaj7 -- C, E, G, B

C6 -- C, E, G, A