A modulation occurs when a piece is trundling along in a particular key (e.g. D minor), and the composer takes the piece to a different key (e.g. G minor).
The composer can use a variety of devices to move the piece into the new key (cadence, sequence, etc.) and the point at which the piece changes is where it modulates. This may be the bar in which the cadence occurs, or the phrase that contains the sequence.
It's a nice trick to string out the modulation as long as possible, so that the tonal base of the previous section is all but lost, in order to move to a remote key smoothly - but it can be very dramatic to perform a sudden modulation into a remote key.
Moving a piece from one key into another is not modulation - it's transposition, and transposing the key of a piece will change the whole character of a song, with one exception; If you are playing a piece on the piano (or other instrument with fixed pitch ratios) and decide not to play it in Eb, but in D# instead, then it will not take on a new character - it cannot, as the notes are the same.
However, on a violin (or other instrument with variable pitch ratios), as has been pointed out, the sound (and hence) character will be subtly different.