What is Inspire in Hearthstone?
Hearthstone has many card effects that are common enough that they get their own keyword - a word or two that immediately tells you how the card works. Let's take a look at the Inspire keyword.
Does something after you use your Hero Power.
Inspire was the first proper keyword added to the game since launch, debuting in The Grand Tournament, which is heavily thought of as one of the worst sets in the game's history, simply due to many of the cards being underpowered at the time of release. This unfortunately did carry over to the Inspire keyword itself as well, and as a result, hasn't been seen in the game since. Although Phase Stalker and Dragonbane would eventually be printed in Descent of Dragons with better success than the majority of the Inspire cards and behaves the same way, it doesn't use the Inspire keyword.
Inspire allows minions to potentially snowball around if they stick around for multiple turns. In Hearthstone though, minions generally don't stay alive for very many turns. If you wish to use the Inspire effect on the same turn you play the card (which you probably will), then you will essentially need to add 2 to the mana cost. This is particularly a reason why many Inspire cards aren't that good. Despite this though, there are some exceptions and some Inspire cards are quite good.
Inspire is also quite bad in Rogue, where the Hero Power is not intended to be used every turn.
Keywords that Work Great with Inspire
You'll want to protect your Inspire minions to give them a chance to survive another turn. In this case, summon a Taunt minion (or multiple) or give it Divine Shield as ways to help protect them.
Playing Against Inspire in Hearthstone
Most Inspire minions are fairly weak for their cost (made even worse by the Hero Power cost of 2 to activate the effects), so they shouldn't be very hard to take out. The effect are generally not very threatening either, so you can also choose to ignore them.
Example Hearthstone Inspire Cards
Lowly Squire is the first Inspire card we saw and it shows. It's bad, don't run it. Using your Hero Power on turn 2 is not always a bad play, although playing this card on turn 1 will force you to do this if you want the extra Attack.
Coliseum Manager is a fairly questionable use of the mechanic, and also one of the only cards that doesn't benefit from using it multiple times. It's also the only Inspire effect that can be thought of as a drawback. Although there are positive applications for the effect, it's usually going to be bad. When it returns to your hand, you will need to use 3 mana again to play it again. 2/5 is pretty decent for 3 mana, but that doesn't offset the bad aspects enough.
Thunder Bluff Valiant is one of the most successful Inspire cards. In addition to having high-Health making it less risky to play on curve, its ability is also quite strong.
Confessor Paletress is another fairly successful Inspire card. Arguably the most fun one and most powerful as well. Although it usually isn't very hard to kill, if it does stick around for another turn, it can summon another Legendary and all of a sudden you'll gain some pretty hefty value. It is sadly less good now than it was back then since many small Legendaries have been released since. If you want to add some "flair" to your Wild Priest deck, Paletress is a good choice. It can certainly win or lose you the game when you play it.