Just ordinary “kitchen” bridge, no especially odd bidding conventions. Here is the bidding sequence:
South: 1 club
West: 1 heart
North: 2 hearts
What does the 2 hearts bid mean?
If you’re entirely sure it has no specific conventional meaning, I would interpret it as something akin to a takeout double, but promising a hearts stopper, and probably denying a particularly long spade suit. So saying something like “we should probably end up in 5C or 3NT, maybe with an offside shot at a slam”.
Since a new suit is forcing, this means something specific: a very good hand with club support and a control in hearts.
e.g.:
S: KJx
H: x
D: Axxx
C: AJTxx
So the heart stopper might take the form of Ax or Kx, or a singleton or void? But probably not a strong heart suit like AJTx or something?
Look up “cue bid limit raise.” Under that convention, the bidding here would suggest 10-11 points and strength in the bid clubs.
It’s a cue bid. Says nothing about hearts. It’s a strong ‘tell me more about your hand’ bid. You may infer that the bidder does not have 4+ spades.
Well for starters, it’s forcing. You can’t pass. 
Seems obvious, but that’s about it for “kitchen bridge” - anything else is a more specific reason for wanting to make sure you can’t pass that would require some discussion with partner.
While you wait to find out why partner wanted to make you bid again, consider your first responsibility to show a heart stopper or extra length in clubs.
Many people in the duplicate bridge world would use this 1m-(1M)-2M cue bid to include invitational hands (about 10-12 points) that could pass 2NT if you bid it to show a heart stopper and a minimum opener, meaning if you held a better hand, you’d have to jump or bid 3M.
No, I wouldn’t assume the 2H cue bid shows good hearts. Partner may actually have good hearts, sure, but he’s not promising it. A cue bid generally takes control of the auction, asking (forcing) partner to “say something more about your hand”, and what the cue bidder does next will tell you why.
The most likely reason to cue bid here is to find out if you can make 3NT, if YOU have a stopper. With a good hand and hearts stopped, partner would be able to just bid 3NT without pussyfooting.
Unless 3NT isn’t what he’s exploring, but it’s the likeliest reason. If he’s looking for a club slam, he’ll bid like a man (or woman) looking for a club slam after you bid 3NT, by bidding on past 3NT.
Upon reconsideration, Quartz has the better answer. My hand would be suitable for an intervening preempt: 1C, 3H, 4H