There are two "new" units for Terran. I use the term new loosely because one is simply a transformed version of an existing unit.
The widowmine is a very powerful unit and can be used both offensively and defensively. As a defensive unit, it's wonderful for warding off drops. One or two can effectively protect your mineral line for other terran drop harassment. You'll want to place the mine that enemy forces will be coming into. If you've ever used vulture mines in Starcraft 1, it's the same mentality for defense.
As an offensive unit, the power that the mine possesses is very good against any unit as it is essentially un-blockable damage. The damage it does is spell damage, so much like a psi-storm, it does full damage on contact. Again, it's all about placement.
The mine itself does not have a lot of health. You want the mines to be in place before your attacks/defense needs to occur. The top of ramps, 1/2 a screen behind the battle area so that when you "retreat" the enemy gets drawn into it, even in lower leagues, you may be able to sneak a drop in and activate a mine on your opponents mineral line fully decimating their harvester count. These are all good places to use the mine.
The biggest benefit to using the hellbat is that in its hellbat form it is considered a biological unit. What does that mean? It means that they can be healed by medivacs! Hellbats also gain a 50% health buff in this form, but are a slower moving unit and lose their long splash damage in favour of a cone effect in front of them.
I'm only going to address how to use the hellbat as that is how the question is formed. Yes there are methods that you can say use the hellion to harass and for map control and then use the hellbat later on for other purposes.
How to use them really depends on your game play style. However, the general consensus is that they are used for tanking damage while other units deal damage to the opposing forces. Depending on if you are going bio or mech the hellbats are in front while the marines/marauders/tanks are behind them dealing damage to the opposing forces.
This early air unit allows protoss players versatile playstyles early on in the game while providing excellent defense for your home base and the ability to detect cloaked units (casted spell).
The Photon Overcharge Ability allows your Nexus to become a defensive attacking structure with excellent range (20 damage and 13 range). This is useful for protecting your base from early attacks and protecting your probes.
The Time Warp ability is a casted area of effect that slows all units caught within it. It reduces movement by 50%. Excellent for chasing down units trying to escape. A few sentries combined with this effect allows chargelots to close in, and decimate an army. Attack speed is not reported to have been slowed by this spell, only movement at this point.
It also has the Mass Recall ability of the original Mothership in Wings of Liberty. As such, this can be use for both offensive and defensive capabilities. Your army is far away and you're getting attacked at home? 2 seconds later and your army is home (assuming your Mothership core didn't get killed).
It also upgrades into a Heart of the Swarm version of the Mothership, which compared to the Wings of Liberty version, has Time Warp instead of Vortex.
The Tempest is a long ranged siege ship, mostly in an offensive capability. It has a massive attack range and does heavy damage against massive air units (Carriers, Battle cruisers, Brood lords, etc.). Again this is a slow unit. The usage behind this would be much like using brood lords or siege tanks (or guardians from Brood war if you want to go back farther). They have a slow fire rate as well so you want to ensure they are protected while they rain down damage from the skies. The biggest benefit of this unit is that it can hit both air and ground. Because of its massive range the unit can attack farther than it can see. You will need other units leading the way to ensure this unit has maximum benefit.
This is a very fast, offensive unit that is frequently used to grant high ground vision and hit and run tactics. It also has a support ability to slow units to a crawl so the ground forces can catch up.
The unit itself doesn't have a direct attack, but you can use its pulsar beam when it has energy to do so. When it is activated, it does fast, high amounts of damage to light units. This makes them ideal for charging into harvester lines and decimating the income of your opponent (un-upgraded 2 shots will kill any harvester). With enough energy, you can send in a few of these units, time warp the harvester line and activate the pulsar beams to destroy their economy while you race ahead.
They also have a Revelation ability that binds itself to enemy units/buildings and will constantly provide vision until the timer runs out (60 seconds) and does not provide detection.
Envision is a self casted ability that grants the Oracle detection for 60 seconds.
This is the new defiler if you ever played Starcraft:Brood War. It is a support unit and has no direct attacking abilities.
Blinding Cloud completely negates the range attack of all units forcing them to become melee fighters. This ability used against any army can be devestating if there is no melee backup. A cloud on a group of marines and marauders makes them useless against a zergling army. It provides time for those lings or ultralisks to cover the ground between the units without taking any damage. Well placed clouds can cripple the power of an attacking army while your melee units do damage (roaches melee when close enough).
They also have an ability called Abduct which does pretty much what the name says. It allows you to pull a unit from where it is to where the Viper is. The cast range is very long (9, same as extended thermal lance of a colossus) so you can pull fortified units to an exposed position where they can be destroyed. Think of "charging" into some marines with tanks behind them with your army only to have them retreat and hide behind the tank fire. With vipers you can pull those tanks forward as you're charging and those tanks then become the first things targeted as the marines end up running behind the tanks because they were pulled forward.
All these abilities take energy, which can be replenished from zerg structures at the cost of 200 HP for 50 energy which takes 20 seconds to convert in full.
Think of these like a burrowed siege tank that spawns ranged broodlings (called Locusts). They are an offensive unit that can be hidden behind mineral lines, or used to provide a first wave to absorb siege tank fire before your main army charges in. The usage of them is no different than spawning infested terrans from an infestor except there is no energy cost, only time. Enemies require detection in order to attack the Hosts when they are burrowed and spawning Locusts so you could get them into the front lines early, spawn locusts, and just keep them there. While this isn't very safe it can keep your enemy at bay for a while. I believe these can be used similar to Widowmines in both the offensive and defensive manner described previously.