A diminished seventh chord consists of a stack of minor thirds, or a diminished triad with an additional third on top.
The diminished seventh between the chord's root and seventh gives it its name. (It is often called a fully diminished seventh to distinguish it from the half diminished seventh.) Note that this seventh can be respelled as a major sixth, thus changing the implied root of the chord:
Dominant seventh is built on the 5th degree of the scale (the dominant) and consists of the triad plus the 7th note from the root.
So in C the dominant seventh is G B D(the triad) plus F (7 notes up from G). It is F natural because it's the dominant of C and not of G.