Mining is the process of adding transaction records to a coin’s public ledger of past transactions. This ledger of past transactions is called the blockchain as it is a chain of blocks. The blockchain serves to confirm transactions to the rest of the network as having taken place. Nodes use the blockchain to distinguish legitimate coin transactions from attempts to re-spend coins that have already been spent.
There are many ways to mine for a chain/coin, it depends on what the developers designed their blockchain for. Some examples are proof of work, proof of stake, and proof of burn. They are explained a little below.
Proof of work is a piece of data which is difficult (costly, time-consuming) to produce but easy for others to verify and which satisfies certain requirements. Producing a proof of work can be a random process with low probability so that a lot of trial and error is required on average before a valid proof of work is generated. Bitcoin uses the Hashcash proof of work system. Hashcash proofs of work are used in Bitcoin for block generation. In order for a block to be accepted by network participants, miners must complete a proof of work which covers all of the data in the block. The difficulty of this work is adjusted so as to limit the rate at which new blocks can be generated by the network to one every 10 minutes. Due to the very low probability of successful generation, this makes it unpredictable which worker computer in the network will be able to generate the next block. For a block to be valid it must hash to a value less than the current target; this means that each block indicates that work has been done generating it. Each block contains the hash of the preceding block, thus each block has a chain of blocks that together contain a large amount of work. Changing a block (which can only be done by making a new block containing the same predecessor) requires regenerating all successors and redoing the work they contain. This protects the blockchain from tampering.
Proof-of-stake (PoS) is a type of algorithm by which a cryptocurrency blockchain network aims to achieve distributed consensus. Unlike proof-of-work (PoW) based cryptocurrencies (such as bitcoin), where the algorithm rewards participants who solve complicated cryptographical puzzles in order to validate transactions and create new blocks, in PoS-based cryptocurrencies the creator of the next block is chosen in a deterministic (pseudo-random) way, and the chance that an account is chosen depends on its wealth (i.e. the stake). In PoS cryptocurrencies the blocks are usually said to be forged (in the blacksmith sense of this word), or minted, rather than mined. Also, usually all the coins are created in the beginning and the total number of coins never changes afterwards (although there are some other versions of PoS where new coins can be created). Therefore, in the basic version of PoS there are no block rewards (e.g. as in bitcoin); so, the forgers take only the transaction fees.
Proof of burn is a method for distributed consensus and an alternative to Proof of Work and Proof of Stake. It can also be used for bootstrapping one cryptocurrency off of another. The idea is that miners should show proof that they burned some coins - that is, sent them to a verifiably unspendable address. This is expensive from their individual point of view, just like proof of work; but it consumes no resources other than the burned underlying asset. To date, all proof of burn cryptocurrencies work by burning proof-of-work-mined cryptocurrencies, so the ultimate source of scarcity remains the proof-of-work-mined "fuel".