I just played the first 2 tutorials of the DS version. It's not really like Advance Wars past the the fact that they're both grid-based strategy.
The way you have your turns is interesting.
When you start, it is your "Move Phase". You move your units.
Then it is your attack phase. You select units and tell them what to attack. Then it become your move phase again, and so on.
However, this is where it gets interesting. When it is your move phase, it is your opponent's attack phase, and vice-versa.
When you make your moves, they don't take place straight away. Instead you end your phase and THEN your moves take place. Your opponent does the same with attacks. Attacks take place before moves, so when everything moves your units are attacked and then they move. The opposite happens with your attack phase.
This means that when you choose where your units will move, you have to think of what units will be attacked, and choose their position based on that.
For example, you might move a unit to an airport to heal because you think it might be crippled by the attack.
It's pretty fun.