The engagement party is a time when the families of the couple will get to know one another. In some cases, this might be the first meeting between the two families or groups of friends and any icebreaker activity will be a welcome event.
In that light, whoever plans the engagement party (likely the bride's family, but it can be the engaged couple or anyone else who wants to plan the party) should plan a few games and activities designed to help everyone get to know everyone else.
First up is a trivia game. Create a "Trivial Pursuit" type game with questions about the bride and groom's lives. You might contain the questions to just facts and events relating to both the bride and groom (such as how long did it take her to say "yes" when he asked, where did he propose, where did they meet, etc), or you can include questions pertaining to their lives outside of each other and before they met each other. Not only can this be fun, but also it's an entertaining way for people to get to know each other and the engaged couple better.
One popular icebreaker that's used at corporate functions and company parties can also work really well at engagement parties. Tape a card to each person's back and encourage him or her to work the room, mingle with everyone and particularly try to get to know someone they have never met before. Before moving on to someone else, be sure to make a comment about the person on the card on his or her back. Partiers write an impression of that person, such as "she seems sweet" or "he knows a lot about the weather".
Toward the end of the evening, the cards are flipped over and the number side is shown. Everyone gets a piece of paper and writes the numbers on the paper, then tries to correspond the name of someone with their number. This fun game can be hard for people who are bad with names, but it's fun nonetheless.
For an activity that doesn't put people on the spot quite so much, consider letting the already marrieds help out the to-be marrieds. Place two pieces of posterboard on the wall and mark them "advice from women" and "advice from men". Now is the time to offer advice about wedding planning, not about being married. That advice can come later. Encourage guests to offer their own wedding planning advice. The advice from older people at the party could be decidedly different from the younger couples in the group, making for an enlightening group of comments.