The halving refers to the mining reward.   

Basically, every 10 minutes (approximately) a new block is mined, the miner who find the block gets a reward of bitcoins (on top of all the fees of the transactions).

Originally, this mining reward was 50 bitcoin. But every 4 years (approximately) the reward will half. The expectation is that by the time the rewards become low, bitcoin will be used enough so that the transaction fees alone would be worth much more than the block rewards, so that miners still have an incentive to keep mining.         

An expected effect of the reward halving is that bitcoin becomes more valuable, because mining becomes less profitable, so miners either need to sell their bitcoins for a higher price to make more profit (because for the same amount of energy spent, they now get roughly half the bitcoin), get more efficient miners or shut down their business.     

Since miners have a large effect on the price (because most miners sell a lot of bitcoin to keep paying their electricity bills) if miners are forced to sell for higher prices to remain profitable, this should drive the price up.     

As long as miners still mine a relatively large % of bitcoin (relative to the total amount in circulation) this effect should be pretty large, but the effect will get smaller once nearly all bitcoins have been mined.