What is the genesis block? In which ways is it important or different from other blocks?
It is important, because it is used to seed the block and chain databases before syncing can occur (you need a block to build on). It's the root of the tree of blocks, forks, and ultimately the chain tips that represent the chains your software is tracking.
It is imported differently to others, as it's coinbase transaction is not included in the UTXO set (it's unavailable to be spent)
Changing the genesis block is a way to definitively fork yourself away from the bitcoin blockchain, ie, start a new network with it's own separate history.
However, some coins do share a common genesis block (Fluttercoin IIRC, shares Litecoins for whatever reason), but the network is forked by the use of checkpoints (genesis might be the same, but block 1 is checkpointed to Fluttercoins, instead of Litecoins).
The genesis block is the first block in a blockchain. Every blockchain based crypto currency must have a blockchain, which is the ledger for the currency. For example, Litecoin has it's own genesis block which is completely different than the Bitcoin genesis block.
Every block in a blockchain has a parent block, except the genesis block which has no parent block.