What is the "confirmation" field in a block? What is the importance of the "confirmation" field in a block?

How many confirmations are required to accept a block in a blockchain?
Blocks in Bitcoin, as they exist in the blockchain, don't actually contain a confirmation field. When you query for a block in bitcoin-rpc or similar, additional information is added to the block based on your client's knowledge of the current state of the blockchain.

A pure block contains only five fields (some of which have sub-fields):


Magic number (to identify that the following info is a block)
Blocksize (how many bytes to read until the end of the block)
Block header (important for block ordering/verification)


Version (the "rules" that this block adheres to)
Previous Block Hash (a fingerprint of the block this block builds on)
Transaction Merkle Root (a fingerprint of all of the block transactions)
Timestamp (UNIX epoch)
"nBits" or a compact expression of the block's PoW target
Nonce (a number the block miner used to find a valid PoW solution)

Transaction count (number of transactions in block)
Transaction list


Back to your question, the "block confirmation" count of a block is simply how many other blocks build on top of that block. Take for example the following three-block-long blockchain:



The blue block (which starts the blockchain, so it's also called the genesis block) has three confirmations (it confirms itself, plus the two blocks that build on it further reinforce it as legitimate). The green block has two confirmations (it confirms itself, plus the one block that builds on it further reinforces it as legitimate). The orange block only confirms itself, giving it one confirmation.

If another block were added to the blockchain:



All of the blocks from the previous image (blue, green, orange) would receive one additional confirmation, since they've been further "buried" in the blockchain.

As soon as a block exists (so it confirms itself automatically, giving it one confirmation) it can be included into the blockchain. In the event of a network fork, however, this isn't always the case. The number of confirmations of a block indicates how difficult that block would be to overwrite. 

In the length-4 blockchain shown immediately above, attempting to overwrite the red block would only require introducing a block which competes with red as a child of orange:



Note that introducing the black block into the blockchain wouldn't necessarily overwrite the red block. If miners choose to mine on top of the black block instead of the red block, making the blockchain [blue-green-orange-black-...] the "longest" (really "heaviest," see bottom of post) chain, then the black block will replace the red block. If miners instead choose to mine on top of the red block, making the blockchain [blue-green-orange-red-...] the longest chain, then the red block won't be replaced. To ensure (assuming network miners don't explicitly favor the red block for some reason, and mine on top of it even when it isn't the latest block in the blockchain) that the red block gets forked, an attacker would need to introduce two blocks which fork off of orange:



To fork the green block off of the network, however, an attacker would need to introduce at least three blocks which build an alternative blockchain off of the blue block:



And for a much higher chance of a successful attack, would need to introduce four blocks which build an alternative blockchain off of the blue block:



Note that this attack would have (possibly-intended) side-effect of also removing the orange and red blocks from the blockchain.

Earlier, I used the term "longer" when referring to competing blockchain forks. While in reality the chain with the longest length measured in blocks is almost always the best blockchain, it isn't necessarily. Rather than count the number of blocks on the blockchain, bitcoin nodes choose the blockchain which is the "heaviest" or has the highest cumulative difficulty. Bitcoin mining involves "brute-forcing" a solution to a problem by repeatedly performing double-SHA256 calculations on a potential block header (described briefly above) searching for a resultant hash with certain properties (which can be incorrectly but essentially summarized as "a lot of leading zeroes"). A higher difficulty means that, on average, miners will have to perform more work for each valid PoW solution. The "best" or "heaviest" chain is the chain which has the highest total difficulty (which is calculated by adding up the difficulty of each block).

To summarize, a block can be accepted to the blockchain as soon as it exists, and it has one confirmation by simply existing. However, blocks of arbitrary depth can be removed from the blockchain by a blockchain fork.