The "key" governs the individual tones of the scale, which in turn will determine whether a chord is "major" or "minor." For example, in D minor the B is flat, and all the other notes are natural. Thus, the D minor chord is D-F natural-A, a minor chord. The second chord is E-G-B flat, a diminished chord. The third chord is F-A-C, a major chord. So, to answer one of your questions, chords in each key can be major or minor (or diminished, which can be thought of as similar to minor). Here's an exercise for you: go through the entire scale, for any particular key, and build the chord on each step of the scale. This exercise will give you a basic understanding of how each chord is built in a key. Then do this in another key, and then another...eventually you will build every chord in every key. You will find that the patterns are exactly the same throughout all the keys.
Now I must complicate things a bit more. Minor keys often "borrow" the V (five) chord (the chord built on the fifth step of the scale - also called the "dominant") from the major key of the same name. Thus, D minor would borrow the A major chord from the key of D major. The theory is that the V chord needs to give the distinct feeling of wanting to resolve to the "tonic" (the first chord of the key - in this case D minor). The dominant being a major chord gives this feeling or tendency. Bottom line for you is that you will need the I (one or "tonic" chord), the IV (four or "sub-dominant" chord), and the V (five or "dominant" chord) in virtually everything you do, so learn those first and memorize them. In major keys, the I, IV, and V are major chords. In minor keys, those chords are minor - but we "borrow" or "substitute" the major V chord from the major key. Good luck - I hope this helps a bit!