The Min Weight Magnitude determines how 'hard' the POW puzzle is. How does this work? What are the implications when setting a lower or higher MWM, how does it impact the consense and are tx with different MWM compatible?
IOTA's proof-of-work is formed by successively hashing the data in a transaction, including a nonce, and increasing the nonce until producing a hash which ends with a particular number of zeroes. This is hard to calculate, but easy to verify given the nonce. Here you can see the weightMagnitude of a transaction is simply a calculation of the number of trailing zeroes in its hash. Three zeroes equals a 9-tryte, which is why IOTA transaction hashes end in several 9s.
The minWeightMagnitude is the minimum number of zeroes that a proof-of-work output/transaction hash must end with to be considered valid by full nodes. Increasing it by one will mean it takes roughly an order of magnitude longer to produce a valid hash. This will make it more difficult to attack the network, but also more difficult for a legitimate device or user to produce a valid hash in time.
Transactions with different weight magnitudes are compatible. This can happen when the device doing POW actually produces a hash with more trailing zeroes than necessary. As long as it is above the minimum, it will be accepted by validating nodes.