There's two phases to scanning for War Assets:
On the system map, you can scan a region of the system by sending out something that could be compared to a "sonar ping," and once you've located an item on a planet or moon, you can select that planet or moon and begin the planetary scan.
In the first phase, you press a button to scan a region around the Normandy. Any items of interest will be highlighted with a red ring. You can then go and investigate these sites. You can only scan a portion of the system at a time, and scanning too many times in the same system (if is is under Reaper control) will lead to the Reapers chasing you out of it, lest you game over.
Some items you will just find in the system map - items like extra fuel from the remains of a wrecked ship, for example. Others will require you to scan the planet before you can locate the item.
If you must scan a planet or moon, you enter the second phase. The second phase is very similar to the way mineral/anomaly scanning worked in Mass Effect 2. You'll be shown a 3-D model of the planet, and you'll have a little radar icon that shows you which direction the item in question is. You'll have to move over the surface of the planet, scanning periodically to keep moving in the correct direction. Once you've located the item (noted with a white dot) you can launch a probe. The probe will recover the item, and then you can leave orbit.
The scanning is quite a bit more straightforward and fast compared to ME2. In that game, you might spend 10 minutes extracting resources from a planet, and use up probes that had to be replenished. In this game, there's usually far fewer items to be found on a planet, and you have infinite probes.