Even among the chords that naturally occur in a key signature, some are much more likely to be used than others. In most music, the most common chord is I. In Western music (Section 2.8), I is the Tonal Center of the music, the chord that feels like the "home base" of the music.
As the other two major chords in the key, IV and V are also likely to be very common. In fact, the most common added-note chord in most types of Western music is a V chord (Naming Chords Within a Key) with a minor seventh (Major and Minor Intervals) added (V7). It is so common that this particular flavor of seventh (Section 5.4.3) (a major chord with a minor seventh added) is often called a dominant seventh, regardless of whether the chord is being used as the V (the dominant) of the key. Whereas the I chord feels most strongly "at home", V7 gives the strongest feeling of "time to head home now". This is very useful for giving music a satisfying ending. Although it is much less common than the V7, the diminished vii chord (often with a diminished seventh added), is considered to be a harmonically unstable chord that strongly wants to resolve to I. Listen to these very short progressions and see how strongly each suggests that you must be in the key of C:
C (major) chord(I)41; F chord to C chord (IV - I)42; G chord to C chord (V - I)43; G seventh chord to C chord (V7 - I)44; B diminished seventh chord to C chord (viidim7 - I)45 (Please see Cadence for more on this subject.)
Many folk songs and other simple tunes can be accompanied using only the I, IV and V (or V7) chords of a key, a fact greatly appreciated by many beginning guitar players. Look at some chord progressions from real music.
A lot of folk music, blues, rock, marches, and even some classical music is based on simple chord progressions, but of course there is plenty of music that has more complicated harmonies. Pop and jazz in particular often include many chords with added or altered notes.
Classical music also tends to use more complex chords in greater variety, and is very likely to use chords that are not in the key.
Extensive study and practice are needed to be able to identify and understand these more complex progressions. It is not uncommon to find college-level music theory courses that are largely devoted to harmonic analysis and its relationship to musical forms. This course will go no further than to encourage you to develop a basic understanding of what harmonic analysis is about.