The AHL's website states that most players are signed with NHL contracts, but the AHL and NHL are not the same. Given so, how does it make sense that a non-NHL AHL player would normally have to be signed on an NHL contract, if one who is playing for the AHL is not an NHL player?
Under section 11 of the NHL Standard Player's Contract (exhibit 1 in the 2013 CBA, starting at page 310),
It is mutually agreed that the Club shall have the right to Assign or to Loan this SPC [i.e. Standard Player's Contract] and the Player agrees to accept and be bound by such Assignment or Loan, and will faithfully perform and carry out this SPC with the same purpose and effect as if it had been entered into by the Player and such other club.
It is further mutually agreed that in the event that this SPC is Assigned, or the Player's services are Loaned, to another club, the club shall by notice in writing delivered to the Player advise the Player of the name and address of the club to which he has been Assigned or Loaned, and specify the time and place of reporting. If the Player fails to report to such other club, he may be suspended by such other club and no Paragraph 1 Salary shall be payable to him during the period of such suspension.
In other words: Hockey teams are allowed to loan their players. The CBA sets rules on when, how, and to who players can be loaned, but part of the contract a player signs is "we can tell you to play hockey for this other team and you have to then go play hockey for that team." The NHL team decides the player is best employed playing hockey for an AHL affiliate (or other non-NHL team), so his assignment is "play hockey for them."
This isn't that unusual a concept, even in non-sports contexts. There's no reason the work your employer assigns to you can't be "go to this other company and do work for them;" this is exactly how temp agencies work.
AHL teams are all NHL affiliates. If you play for an AHL team, you're playing for an NHL affiliate team.
In other words, most players who play in the NHL are draft picks -- and they are signed with an NHL contract but to play for an AHL team. Again, this means a player may be moved up or moved down between AHL and NHL teams, and this happens very often. AHL players tend to get paid significantly less than NHL players -- especially if a rookie player or the like, since the AHL is more second-tier.
It is possible to play for the AHL without being a draft pick though -- sometimes teams have open tryouts and scout free agents, but this is less common than the latter.
To summarize though, a player will sign a contract regardless of how he gets accepted to a team -- and the contract is going to be pretty much the same difference whether a draft or scout pick.
There are all kind of different contracts you can get in the NHL. Usually, a good player that statistically will make the team at the beginning of the season would sign a one-way contract for the respective NHL team. If this player gets cut from the team or lower to the AHL, he has to go through 'waivers' before he can get assigned to the AHL. If another team picks him, he will have his NHL contract moved to that team and he will now belong to that team. If no team claims him off the waivers, he will be assigned to the AHL and given an AHL contract.
Also, players sign two-way contracts which means that if they do not make the NHL team, they will have a contract in the AHL already set and planned. Most 4th line players or AHL players will have a two-way contract.