In this article we shall consider the sequence beginning 1♥-(1NT). This is a common enough sequence, and one which many players find hard to deal with, in spite of its relative simplicity.
1♥-(1NT)-X: No surprises here. This shows that you think they are not going to make 1NT. Typically you might do this whenever you have about 8 or more points. The exact number you require will depend on how aggressive you are, and how accurate your defence is! But conceding 1NT doubled is not such a disaster, so don't worry if the odd ones here and there make.
1♥-(1NT)-2♣/♦/♠: Weak non-forcing. You haven't got enough to double and are trying to compete the part score. Should be a six card suit, and partner should normally pass it, only pulling to his own suit with a good six card suit of his own, and little support for partner.
So far so good? This is most of what you need to know about this sequence. Of course you can also raise partner, but aside from that most hand types seem to have been catered for pretty well. If you follow the above you won't go far wrong, but the big worry some people have is what to do when they have enough strength to double but too much distribution.
The simple answer is to say that these hands don't exist, and carry on as before, but this is not the end of the story. The two sequences undiscussed so far are three level bids, and the use of 2NT. The latter is an easy tale to tell - the traditional use is for a big two suited hand (possibly including a weak hand with the minors) - at least 5-5 in two suits. (Close your ears whilst I mention that some people like to combine a good raise in hearts into this.)
1♥-(1NT)-3♣: Much of this is a matter of opinion. This could be preemptive in clubs, strong with lots of clubs, or showing a heart fit and clubs. If you generally play fit jumps then I feel that the last meaning is the most useful. If not, preemptive is probably the best meaning, sticking to the philosophy that you will double with a strong hand. Of course, you are not really expecting opponents to stand 1NT doubled, and you hope to be able to show the suit later.