A share is a proof that your mining software is working hard and working correctly.  It's the same thing as submitting a solved block to the bitcoin/litecoin network, but it's done with a lower difficulty setting.  This means your mining software can send frequent shares to the pool server, and the pool server can check they're valid and tell how hard and effective your miner is.  If any of your shares are actually strong enough that they do solve the block then the pool server passes them on to the network and usually collects the reward itself, then distributes it to the miners according to share levels, by various algorithms.

I find the dice metaphor really useful.  Imagine Max and Megan are rolling 100-sided dice, and Naomi will pay them $10 every time they roll a 1.  That's pretty boring, it could take a long - and variable - amount of time to get any money.  So Peter sits between them and Naomi, and offers to pay them $1 every time they roll a number 9 or lower - and if they roll a 1, he passes the die to Naomi and she pays Peter $10.  On average this almost balances out - Peter takes a slight cut, and also has to loan the money some of the time.  And it's more fun and practical for Max and Megan because they get "hits" more often, and they get paid more often, even if the payment is in smaller amounts.

In this scheme, Naomi is the bitcoin/litecoin Network, Peter is a Pool server, and Max and Megan are Miners.  Rolling dice is hashing blocks, the 1-rolls are solved blocks, and the 1-to-9-rolls are shares.  Sometimes the shares also count as completed blocks, but rarely.  Peter gets to choose the hit rate for his miners.  Maybe Paul is another pool server, with a different hit rate, payment scheme, fee, or membership.  The miners can choose which pool to use based on those criteria.

So to understand why hash rate may be high but share rate low, suppose Max buys some extra dice, and rolls three at a time - then he can win more often, as his hash rate is higher.  But if he tries to roll five dice at once, or if he rolls the dice too quickly, then some of them fall off the table and onto the floor.  In this case they don't count, so although he's rolling the dice really fast, he can't submit his shares to Peter - or, he submits them and then Peter says that they don't count.  This is analogous to tweaking your mining settings to values that appear to give a good hash rate, but in fact lead to a poor share rate.

It's also possible that it's best for him to roll 20 dice at once, and have 15 of them fall on the floor, rather than roll just three dice and be sure they're all valid - because at the end of the day, even though he wasted a lot of dice, he still got a higher share rate overall.  It comes at the expense of the hassle of rolling more dice, of owning more dice, and the wear and tear on the dice.

I know that's a bit of a tangent but I think this metaphor is very useful for intuitively understanding the mining and pooling processes.