The first thing you need to avoid is player elimination. Knocking players out of the game is generally a bad thing, you can only really get away with this if it is both necessary to the game, and if the game is short enough that players don't mind waiting for the next round.
The next obstacle is logical elimination. This is where players can fall so far behind that they have no chance of winning. Even if they're technically in the game, if they have no hope of victory, they might as well not be playing. You need to allow players to have some method of catching up and making comebacks so that everyone can stay engaged. This is sometimes referred to as "rubberbanding"; where a player who gets too far away from the pack gets snapped back towards the median.
There are two forms of rubberbanding: you can either give people that have fallen behind an advantage, or you can impose a penalty on anyone who gets too far ahead. The latter method, I like to call "Blue Shelling" and I generally dislike it - it seems fundamentally wrong to punish the players for doing well.
Comeback Mechanisms should be a subtle thing, as you generally don't want to reward players for losing so much that they would prefer to be in last place. You generally want to avoid just handing them points, and instead give them small advantages that let them get their game plan back on track.
Once you have a game that allows all players to be strategically engaged (by not eliminating any players, logically or otherwise), you need to ensure that players can be engaged by the mechanics themselves. One of the worst offenders for this is long turns that only involve the current player. Players zone out, waiting for their own turn, and then take extra time to figure out what happened while they weren't paying attention.
There are a lot of different ways to combat this, such as:
- The simplest method is to just have turns short enough to keep players from zoning out. If long turns are inevitable or unavoidable, consider reducing player count, so that it at least gets back around faster.
- Having multiple players playing simultaneously. There are a variety of ways to do this; Scattergories, Bananagrams, and 7 Wonders are all good examples of simultaneous play.
- Giving people things to do during their off-turns.
- In Risk, you roll defender dice when you are being attacked
- Puerto Rico has everyone participate in whatever action is being taken
- Kingdom Builder gives players enough information to start planning their next turn before it happens.
- Taboo has a player paying attention and buzzing out players who use any of the "taboo" words.
- Even simple tasks can keep players engaged, giving players "busy work" can sometimes actually make it more engaging and fun.
- Allowing player interaction such as Trade or other negotiations
- Making whatever action the player is taking inherently interesting enough to keep players' attention. The most obvious example would be from games like Cranium or Charades, where players actually perform and do things.
- Making paying attention to the opponent's moves inherently important. A game like Mascarade is an excellent example, as if you don't pay attention, you'll have no idea what is going on.
All in all, there are a lot of things to consider to maximize player engagement. You not only have to keep them strategically engaged by letting everyone have a chance at victory, but also focussed on the game at hand, even when it isn't their turn.