I open 1D with a balanced hand of 19HCP. Partner bids 1NT. No interference from oponents. How do I show my strength in NT in rebid? Should I bid 2NT or 3 NT?
Assuming a relatively standard framework, then partner's response shows 6-9 points with no 4-card Major and an inability to make a weak raise in diamonds. It is generally desired to play in a 3NT game with two balanced hands when the partnership assets total 25 (or a good-matching and equally divided 24) points. Partner's announced 6 with your 19 make 25, so the book bid is a direct raise to 3NT, as stated by ruds.
However, there are strategic considerations. There may be difficulties with so few partnership assets hidden. The opponents may well be playing double dummy from trick 2 or 3 on if a good lead is found. I would tend to underbid by a point here, and raise just to 2NT with exactly 19, when not vulnerable. It is winning bridge to stretch slightly (to game, not slam) when vulnerable, and retract slightly (again wrt to game only - small-slam bidding is reversed) when not vulnerable. This reflects the unequal rewards and benefits of vulnerable and not vulnerable game (or rubber) bonuses.
The raise to just 2NT invites partner to continue on with 8-9 points, and to pass (or correct, if your agreements allow) with just 6-7.
Daily smile for those who know Latin: I saw the use of boni in place of bonuses last weekend in an answer on another site - clearly by someone who didn't know that boni means "good men", and Boni means specifically "the good men who conspired to assassinate Caesar". The meaning of the answer changed dramatically with that interpretation.
You don't mention your bidding system. In Standard American, you would rebid 3NT.
One good option is to reserve the 3NT rebid for excellent 6-card diamond suits. (Just as many partnerships use a 3NT rebid to show 18-19 with 6-card major after opening 1M). So, in this case, you would either rebid 2NT or open 2NT. (In general, I open 2NT for most any 19 count with a 5 card minor, and some other excellent 19s besides.)
Partner bid 1NT (6-9 points balanced), opposite your hand, which was also balanced, but too strong for 1NT. Clearly, you belong in a NT contract, with the decision being between 2NT and 3NT. Your rebid options are between those two bids, and the issue is who gets to decide the level.
If you rebid 2NT, you let partner decide. Your 1D--2NT sequence advertises 18-19 points. (With 20-21, you would have bid 2NT directly). Partner now knows that your strength range is exactly 18-19 and will (probably) give you credit for 18. If s/he has 6-7, partner will probably pass, calculating your combined count at 24-25. If partner has 8-9, partner should raise to 3NT, figuring on 26-27.
If you raise to 3NT, you are taking the decision on your own. This is not recommended. The problem is that you now know the strength of partner's hand only within four points (6-9), whereas after your 2NT bid, partner knows your hand within a two point range (18-19). Better information makes for better decisions.
A 2NT bid (almost) fully informs partner about the strength of your hand (and s/he can make the better decision). This bid makes partner the "captain" in the bidding process.