It is helpful to first have a good understanding of how Blockchain works. This video by 3Blue1Brown is an excellent place to start.
When it comes to comparing IOTA (or more specifically, the Tangle) with Blockchain, there are many differences, as well as many similarities.
A Blockchain is a sequential chain of blocks where each block references its chronological predecessor, similar to a linked list. Blocks contain multiple transactions, and are added in more-or-less regular, discrete time intervals. In the Tangle each transaction (rather than a block of transactions) references two previous transactions, forming not a linked list, but a complex web structure known in mathematics as a Directed Acyclic Graph, or DAG for short. Directed because all the reference pointers point in the same direction, acyclic because you cannot follow the path from any one transaction and arrive back at the same transaction (in other words, no loops), and a graph because the reference pointers and transactions form a graph of edges and vertices. Importantly, this DAG structure allows transactions to be issued simultaneously, asynchronously, and continuously, as opposed to the discrete time intervals and linear expansion of a Blockchain.
By parallelizing transaction issuance and validation, IOTA is able to achieve a significantly higher transaction throughput. In a Blockchain, because many miners are working to add the next block to the chain, with significant overlap between the transactions included in their respective blocks, transactions are forced through a bottleneck by design - this is how the double-spend problem is resolved and how consensus is ultimately achieved. However, the vast majority of transactors are not trying to cheat by double-spending their coins, yet they are forced through this bottleneck nonetheless. Instead IOTA lets all transactions through and cleans up any conflicting transactions using the "heaviest Tangle" rule described below.
One common misunderstanding about the differences between Blockchain and the Tangle is how consensus is achieved. Blockchain achieves consensus through the "longest chain" rule. Miners are incentivized to add the next block to the chain with block rewards and transaction fees. Which miner gets to add the next block is determined by a fair and decentralized lottery-like system known as Proof-of-Work where computational resources must be spent to solve a cryptographic puzzle. The only way to solve this puzzle is by brute force, also known as "guess and check." Because two or more miners may find the answer to this puzzle at almost exactly the same time, thus simultaneously creating new blocks which may contain conflicting transactions, the network needs a consensus-building rule to determine which chain should be accepted as valid. This is where the "longest chain" rule comes in to play.
When a miner sees two valid blocks referencing the same parent block, this is known as a fork. The miner then has to choose onto which side of the fork of the chain to try and add the next block. The probability that miners working on either side of the two competing chains continue to find blocks at roughly the same time decreases exponentially for each subsequent block. Consensus around which chain should be accepted as valid is achieved by saying the "longest chain" wins. However, this means that a transaction cannot be considered confirmed until a significant number of blocks have been built on top of it. The rule of thumb for Bitcoin, for example, is after 6 blocks, or on average 60 minutes, a transaction can be safely considered "confirmed".
Note: The arrows to blocks with conflicting transactions in the Blockchain, or arrows to conflicting transactions in the Tangle, does not mean they are conflicting with eachother in each respective picture, but rather with some other transaction(s) which are part of "true history" on which there is consensus (in teal).
With IOTA, consensus follows the heaviest Tangle rather than the longest chain. In the Tangle, because transaction issuance is tightly coupled with network validation, transaction finality (commonly called "confirmation") can be achieved in a fraction of the time it takes for Blockchain. The more activity there is in the Tangle, the more validations occur, and the faster transaction finality becomes. The vast majority of users in either system, Blockchain or the Tangle, are not actively trying to double-spend or cheat because the majority in either system are assumed to be honest. When double-spends or other forms of cheating do occur, just as in the case with Bitcoin, the 'invalid' branch of the Tangle is ultimately orphaned into oblivion.
With IOTA, to issue a transaction, a tiny amount of Proof-of-Work is required. Unlike Blockchain where Proof-of-Work is employed as a decentralized lottery of sorts, in the Tangle this serves merely as a spam and sybil prevention measure. As more and more transactions occur, more and more cumulative Proof-of-Work is added to the system making it more secure against attacks. The Tangle grows more secure as more activity is added. This is because requiring a tiny amount of Proof-of-Work makes it costly for an attacker to "outpace" the throughput of honest transactions. The cost to issue a transaction for an honest user is inmaterial, but the cost for an attacker to overwhelm the aggregate throughput of honest transactions would be forbiddingly high.
Because there are no miners in the Tangle, and the responsibility for validation is an intrinsic part of issuing a transaction, there are no transaction fees. The value sent is always equal to the value received. This enables feeless micro and even nano-payments which the emerging machine-to-machine sharing economy will require to operate at scale.
In IOTA, every participant in the network making a transaction also actively participates in the consensus. Unlike as it is in the case of Blockchain where there is a bifurcation of roles between the miners and the users of the system whose interests are diometrically opposed (miners want slower transaction confirmation times and higher fees, whereas users want the exact opposite), in the Tangle the incentives of all participants are perfectly aligned.