Background
What bitcoin does is maintaining a distributed ledger in which transactions are stored. Simplified, you will get a huge table like this:
From | To | Amount
------+--------+----------
SYS | A | 100
SYS | B | 100
SYS | C | 100
A | B | 50
C | B | 50
Coins come into the system when they are conjured up out of thin air and transferred to some account holder. The first three transactions in the table above represent A, B and C each receiving 100 coins from the system. From there on, both A and C transfer half of their coins to B. So now we can infer that A and C both have 50 coins, while B has 200.
Double spending and Forks
If C now would try to send 100 coins to A, the transaction should be denied as invalid, because C does not have enough coins. But here comes the tricky part. What if the node processing this new (invalid) transaction had not yet received the last update? In other words, what if it had not yet seen that C transferred 50 coins to B already? It would not be able to say the transaction was invalid and so it would accept it.
This is a fork. The system processing the invalid transaction now has a different ledger from the rest of the world:
From | To | Amount
------+--------+----------
SYS | A | 100
SYS | B | 100
SYS | C | 100
A | B | 50
C | A | 100
In distributed systems, even if no one is dishonest, such forks will happen all the time. Because the time it takes for a transaction to propagate to all the nodes in the network will differ for each node, everyone will have a slightly different view of the order that those transactions happened in. But that order can be critically important, as we saw above. So who is right and how to get everyone to agree on that?
Consensus through Proof of Work
First the question of who is right. Is there even an objective order of transactions? General Relativity suggests that actually, there is not. So which order of transactions we accept is actually not that important, as long as all nodes agree on it.
So how to get them to agree? This is where consensus through Proof of Work comes into play. Each node processing transactions encodes it's view of reality (the order of transactions it thinks is correct) into a block and then solves a cryptographic puzzle, attaching the answer to the block. The role of this Proof of Work is that it acts as a collateral; the processing node had to do work, which means spend real-world money, to create that block. This means it has a stake in that block being accepted. If the block is not accepted, the work will have been done for nothing. This creates an economic incentive to 'do the right thing'. If you do the wrong thing, chances are you will end up having done work for nothing, unless you can get the other nodes to accept your wrong version of reality.
The whole thing is a race. Only one version of reality can be accepted for each block in the chain, so the first miner to solve the puzzle and produce the next block has the biggest chance to win the price. But at the same time, miners receive new blocks from other miners. Multiple blocks with multiple versions of reality. The miner has to pick which of these blocks to accept and use as the base for mining the next block. If it picks wrong, it looses the work invested in mining the wrong chain.
So, accept a wrong block and loose the work done on mining that chain, unless most other nodes also accept that wrong block. Create a wrong block and loose the work done mining that wrong block, unless you get the other miners to accept your wrong block.
Blockchain without miners
I don't think it's possible. 'Miners' is just a term used to describe 'transaction processors'. A blockchain needs processors whichever way you cut it I think.
Blockchain without Proof of Work
It seems to be possible. People are looking at Proof of Stake as an alternative. In that scenario, 'miners' wouldn't need to solve a cryptographic puzzle, but they would need to 'put their money where their mouth is', so to say. Again, this creates a collateral which acts as an economic incentive to do the right thing.
One very subtle thing here is that, although the idea to use the very coin people are mining as the collateral is very intruiging, I'm personally not yet sure whether it wouldn't actually create one big circular reasoning cycle. Because in different forks (different views of reality), the coin may actually have different value... so how to objectively decide what the stake actually is. The subtle genius of Proof of Work is that the work is done in the real world, so the stake is in the one common reality we all share. Meaning it's the same no matter what version of reality the ledger you are looking at contains.
It's a very intruiging question you asked. I hope my answer helps your understanding and I have to add I am not an expert in the field at all so I may have made mistakes.