I often see good players note on their convention cards that 1NT may contain a 6 card minor. Depending on the strength on the 1NT range, I would bid such hands 1m - 2m (12-15) or 1m - 3m (15-18). What are the benefits of opening 1NT with 6 card minor? What are the characteristics of such hands?
This kind of bidding will be popular with pairs that like to open and play a "Gambling 3NT" with a long, strong, minor. The opener will bid 3NT with such, and little else, and the responder will "leave" it there if s/he has stoppers in all three, or at least two out of the three remaining suits. With no such protection, the responder will "pull" to 4 clubs, which may be the final contract, or corrected to 4 diamonds, if that is the opener's suit.
For such pairs, it would logically follow that one would open One No Trump with 15-17 high card points, and a solid six card minor suit, something like Ax Kx Qxx KQJxxx or Ax Ax Kxx KQxxxx. The likelihood is that if partner has a decent responding hand (about 10 high card points), s/he is short, or at least weak in that one minor, has both majors stopped, and perhaps the other minor as well.
With a long, strong minor suit that can take six or more tricks, it is usually easier to get three more tricks for a game in 3NT than five more tricks in a minor suit, which is why many pairs aim for 3NT with such a holding. If one suit is unguarded (hopefully the other minor) the opponents may not lead it, because no trump bidders will often conceal minor suit strength. Since they will usually show major suit strength, the absence of a major suit bid makes it a "moral" certainty that one of the two majors will be led.
The "second" type of 1NT bid with six of a minor is, as ruds pointed out, when you have a long but fairly weak suit, headed by maybe a Q or J, with most of your 15-17 points in the three short suits; something like Ax AK KQx Jxxxxx. Then your hand is actually "balanced" with length substituting for strength in stopping your weak suit. Your hope is that your strong "side" values will find partner with a five card suit that can be run. A Qx (or better) in your weak suit would be ideal, as it would limit the opponents' tricks in that suit.
Holding AKx Kx QJxxxx Kx, it is unsatisfying and misleading to open 1D and rebid 3D. Most of your values are outside of diamonds and in fact you have decent stopping positions in all other suits. Opening 1NT conveys the hand type and playing strength more accurately.
To expand on the answer by ruds, the scoring also plays into the decision.
Where as it requires 10 tricks in a Major suit to make game, and 11 in a Minor, it takes only 9 tricks to contract for game in Notrump. This makes the required total strength required for a good shot at game about 4 Points less in Notrump (~25 combined HCP) than in the minor suit (~29 combined Points).
On distributional hands the additional strength can be available outside the high card strength, and leverage trump length to protect against an opponents length tricks in a side suit. When the hands are balanced, it is usually wiser to play for the reduced total required tricks.
The main advantage of opening 1NT is that it limits the hand in both in terms of strength and distribution. When you open 1 minor, you can have between 3 and 7 cards in the suit, between 10 and 20something HCP and can be void in any suit. When you open 1 NT, your hand is limited in strength and the distribution is well defined. This is a huge advantage if the bidding turns competitive and a large advantage in partnership bidding (just one example: responder, with a 6-card major and game-going strength, can safely place the contract, knowing that opener has at least adequate support).