On this site http://jguitar.com/chordsearch?chordsearch=Am%2FC is said
Am/C to be played like this:
X O ------ ||||O| ||OO|| |O|||| ||||||
but in another tutorial site I saw this
X X O ------ ||||O| |||O|| |O|||| ||||||
Which is correct?
Both are correct. Chords can be played in more than one way.
Am/C means "A minor with C as the lowest note".
A minor: A C E
A minor with C as the lowest note: C A E
The order of A and E doesn't matter on this case, as long as C is the lowest note in the chord. The number of As, Es, and Cs doesn't matter either, again, as long as C is the lowest note in the chord.
The notes on the first example are: C E A C E
The notes on the second example are: C A C E
Both are examples of Am/C. The second one has one less E, but it's still a Am/C chord. It still has A, E, and C; with C being the lowest note.
Which one to use? Whichever you want, whichever sounds better to you.
As JCP says, both are correct. However, one may suit you better to play, or to listen to, or to fit against a particular part of a melody. You'll find, as slim points out, many, many different VOICINGS of the same chord, particularly on guitar. The choice is yours. Context is important, as maybe is the moving from one chord fingering to another. One's ear is probably the best decision maker of all.