Once you have researched the Power Suit and UFO Navigation (in either order, of course), you gain the ability to research the Flying Suit. The Flying Suit is essentially a Power Suit with the added ability to fly. The Flying Suit's protection value is also 10 points stronger per side than the Power Suit, though it requires more cost and materials to produce, but if you have the spare materials and money it's worth it just for the increased protection alone.
The hover unit allows soldiers to raise or lower their elevation at will. This freedom of mobility opens up a wealth of opportunities that you can take advantage of, such as being able to snipe from any position or to scale buildings or UFOs with ease. On the other hand, as the hover unit is designed to keep the soldier in the air, descending while in a Flying Suit is a slightly slower process.
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Flying Suits are great for scouts but can still drop like a lead balloon from one lucky plasma shot. Since there is little cover in the air, be sure to land or take cover so as not to be very exposed during the Alien turn.
Flying Suits are also useful for instantly setting up snipers in high positions anywhere and at any time. For example, you could have them hover over the Skyranger a few turns after deployment to provide long distance cover fire for your scouts as they expand your zone of control. Unlike scouts, snipers generally stay well behind the front lines, allowing them to stay in their positions for as long as necessary.
Melee aliens like the Chryssalid and Reaper cannot attack a levitating unit. If there are melee units in the field - hover one tile off the ground and you will be completely safe from physical attacks. (Warning: Also be sure you are not adjacent to a solid surface that a melee alien can reach. If that Chryssalid does somehow manage to get onto the roof, it can then attack you. Similarly, a grenade deposited on the roof will blow up any nearby soldier.)
Similarly, hovering one tile off the ground can protect you from alien grenades. Since the grenade will land on the level underneath you, it will detonate harmlessly. You could attract grenades this way by huddling together in a group while in the air. Some caution should be taken when doing this as it does make your soldiers one big juicy target for the enemy's ranged explosives and projectiles.
Even while you are hovering, you can still kneel to take better aim and reduce your profile slightly. Your soldier's sprite will not appear to change to reflect this, however all the bonuses and costs involved in kneeling will still be present. You can tell that you are kneeling by the level of the bobbing yellow arrow indicator above your selected soldier's head.