Yes Big imply's major and small imply's minor.
As for witch chord to use (major or minor) in a given key, here is a "quick" explanation.

When you play a song (lets say in the key of C, wich would be the I...big = major :) )
If you want all you chords in the song to be diatonic (diatonic = every chords uses only notes from the given key) your gone have to tweak them a little.
So in the key/scale of C we got :

C (the 1), D(the 2), E(the 3), F(the 4), G(the 5), A(the 6), B(the 7) and back to C(the 8 or 1).

No sharps no flats.
Lets build a chord progression with the 1, 6, 2, and 5 (C, A, D, G)

1- C will be major, offcourse because we are in the C major key.

2- A....We want our A to be diatonic to the key. The 3 notes in the A major chord are A C# E (the 1, the major 3rd and the perfect 5th intervals of A)
But the C scale has no sharps so we are not playing a chord that is diatonic to the key. So we lower the C# to C and that gives us the minor 3rd interval of A. So now we have A minor.

3- D (the 3 notes of D major chord are D, F#, A) we lower the F# to F. Same thing, this gives us a D minor.

4- G (the 3 notes of G major chord are G, B, D) All notes from the G major chord are diatonic to the key of C. So G major is good here.

Our progression would be C Am Dm G.

Here's the "rule" if you want to go quicker, instead of always doing the sequence I just showed you.

For a major key, it will always be like this

I, ii, iii, IV, V, vi, vii or 1=maj, 2=min, 3=min, 4=maj, 5=maj, 6=min, 7=dim

(dim is for diminished)
Youl notice that if we take the 7th note in the key/scale of C (B wich is build with the 3 notes B, D#, F# if its major) and make it diatonic, you will end up lowering the 3rd and the 5th intervals (D# and F# to D and F). Thats a diminished chord.


For a minor key, always like this.

i, ii, III, iv, v, VI, VII or 1=min, 2=dim, 3=maj, 4=min, 5=min, 6=maj, 7=maj

If you want to dig deeper and want to be explained better then what I just tried :) Christopher Schlegel probably as some videos explaining the theory behind all this. He's the go to guy for all your theorical questions. Atleast he's my go to guy and probably is for a lot of people here on GT :)