Checkback is considered to be essential by many partnerships - it is a convenient way of inviting or game forcing on the 2 level. Lets look at how it works.
The standard checkback sequence
In this context, the bids are as follows
2♣ = Artificial (does not show clubs), any invitational hand OR Diamonds and weak.
2♦ = Any game force hand.
Let's look at some examples. I'll start with 2♦ since its a bit simpler, it only has one meaning.
The 2♦ bid
Here we would like to game force with 2♦. Now we can conveniently find a possible major fit, otherwise we will settle for 3NT as the final contract.
NOTE: after the bid of 2♦ (checkback), all bids are natural, for example
Here opener has shown 4 hearts, simple and natural.
The 2♣ bid
a) A hand that wants to play in 2♦.
This is a useful quirk of the system, if you have a hand that suits playing in 2♦, for example
A sensible contract, we used 2♣ checkback just to get opener to bid 2♦, and then passed it!
But make an important note, when responder bids 2♣ checkback, they don't always have diamonds, look at the next example.
b) An invitational hand, typically say 11ish points. For example
We've shown an invitational hand (by going through 2♣ checkback), and have rather naturally just shown spades and hearts. We will be 5 spades and 4 hearts (with 4-4 we would've bid 1♥ initially).
Why bid checkback?
We normally bid checkback to look for a major fit. Don't bid checkback as a matter of routine, only bid it if you need it. For example
Would you bid checkback here? Contrast that to the previous example where we could have a heart or spade fit still, and we would like to investigate
Some important things to clarify
I suggest that checkback only applies after a 1NT rebid. Contrast that, for example with this auction.
In this context I suggest just having 2♣ as natural, it is sensible to be able to play in opener's club suit.
Common issues
1. What happens in competition? My suggestion for most partnerships is to just agree, as always, "System off in competition". This is an easy catch all rule, no one can get confused. So in competition the bids will be natural.
2. Forgetting checkback is very common if you're not used to it. It is important for you and partner to have a good attitude and be patient.
3. Remember it only applies after a 1NT rebid (and not in competition). Don't invent other auctions where it applies unless it is specifically agreed with partner.
Overall I think it is a worthwhile convention to learn. It is high value, high frequency (it comes up a lot), and reasonably easy to remember once you get used to it. Also, you may find your opponents playing it regularly, so its worth knowing what is going on!
Read next: Fourth Suit Forcing: How and When to Use It · Transfers over 1C: How They Work · System Over 1NT: Transfer Responses and Conventions