Offensive zone entry is key to generating good scoring changes. After all, how can you expect to score if you can't control the puck in the offensive zone?
It's not just about whether or not you enter the zone, but also how you do it. Depending on how much support the offense has, and how many defenders are protecting their zone, attacking players have a few different options to get into the offensive zone effectively.
Cross-Ice Dump - This play is more effective when a player has offensive support. The puck carrier will dump the puck into the zone at the blue line, it bounces off the boards about half way between the net and the side boards, and heads in the direction of the offensive support coming down the other side.
Here's an example of Mark Stone banking the puck off the boards with support from Milan Michalek, who would later score.
Rims - The player with the puck senses that he has support, enters the zone and sends the puck all the way around the boards to his teammate on the other side. Instead of the puck bouncing off the boards and heading more towards the middle of the ice, it stays right along the boards. This is often not very effective, as goalies are able to leave the net and stop the puck along the boards before it gets to the other player.
Chips - This involves an individual puck carrier banking the puck off the boards to a space behind the defender. Because the offensive player is skating in the same direction, and the defender is forced to change direction, he can often skate around the defender and get to the puck first. If the puck carrier has support, he can bounce the puck off the boards behind the defender to an area where his fellow attacker can swoop in and pick it up.
Back-Side Pass - The player with the puck carries it into the middle of the offensive zone, and then passes it back towards the space where they just came from to a player with vertical speed. This play shifts the defenders sideways, opening up space for the back-side player.
Ladder Play - In the ladder play, the puck carrier drives across the blue line through the middle and dumps the puck to the outside player, sitting stationary at the blue line. The player at the blue line is left with time and space to shoot the puck, as the original puck carrier drives to the net, dragging the defender with him or screening the goalie.
This play dictates that once the puck carrier crosses the top of the face-off circles in the offensive zone, their only option is to put the puck towards the net. As a result, the supporting offensive players should be getting themselves to the net for a rebound, rather than looking for a pass.
Since the main objective is to get the puck to the net for a rebound, the funnel is most effective when the player driving the initial zone entry is an offensive defenseman, like Erik Karlsson for example. This leaves three forwards available for a rebound, and allows the puck carrier to get back into place as soon as the puck is shot.
When offensive players have a two-on-one opportunity, the key is to maintain speed when entering the zone. If it is a wide odd-man rush, the offensive players should move into the middle as soon as possible.
The puck carrier should get himself into a triple threat position. This is where he has the puck at his side in shooting position, so he can either shoot, pass or a make a move. He then reads the defenseman and goalie to see how they react to the situation. The other player just keeps his stick in a position to shoot or deflect the puck into the net.
Crisscross - In this play, the puck carrier and his teammate crisscross as they enter the offensive zone. The player without the puck crosses behind the player with the puck. This creates lateral movement designed to throw off the defenders. While it can be effective, there is a high risk of going offside here.
Two-on-Two Midlane Drive - As the two offensive players enter the zone, the player without the puck cuts into the middle between the two defenders. As he does this, the puck carrier crosses behind him. This can cause hesitation in the defenders, giving the puck carrier more time with the puck.
High Triangle - As the puck carrier enters, one of his teammates goes wide on the other side. The third attacking player hangs back, just inside the blue line. The puck carrier then has the choice of passing to the player going wide, shooting the puck to leave a rebound for that player, or passing back to the player who is sitting back.
In case you aren't tired of watching this goal yet, which you probably aren't, here is Mark Stone scoring off a (loose) high-triangle entry.
Triple Drive - All three attackers enter the zone. The puck carrier goes wide with lots of speed, both supporting players drive the net, one in the middle and one from the side. As the puck carrier continues to skate outside near the defender, he immediately cuts inside. He can then take a shot from (hopefully) a prime scoring area, or make a variety of plays with the players helping him.
Press and Pull Play - This one is very popular in the NHL. One of the attackers without the puck drives towards the net as soon as the puck crosses the line. Instead of staying in front of the net, this player loops around and finds a soft scoring area (top of the circle or outside the hashmarks). This causes the defender to chase him to the net and, since the defender is reluctant to leave the net, creates separation. The puck carrier then has an easy pass to the player sitting all alone.
Note: The images in this post, and most of the information, are courtesy of NHLGuides.com, where you can find even more zone-entry plays, and other tactical information.