Inversions
A chord is in root position when its root is in the bass. Moving the lowest note up an octave gives the first inversion, then the second (and the third for four-note chords).
Inversions let you move from chord to chord without shifting your hand: for each chord, pick the position closest to the previous one.
Fingerings
Numbers above the keys are right-hand fingers: 1 thumb, 2 index, 3 middle, 4 ring, 5 pinky. For the left hand it is usually reversed: 5 on the lowest note, 1 on the highest.