What is a blockchain?

In short, a blockchain is a public ledger used to record and monitor the transactions of bitcoins. Again, not a central authority, but instead see it as a network of communicating nodes, running bitcoin software.

Transactions of the form payer X sends Y bitcoins to payee Z are broadcast to this particular network, using readily available software applications.
The network “nodes” can then validate transactions, add them to their copy of the ledger, and then broadcast these ledger additions to other nodes.
The blockchain is therefore to be seen as a distributed database – to achieve independent verification of the chain of ownership of any and every bitcoin amount, each network node stores its own copy of the blockchain. Approximately six times per hour, a new group of accepted transactions, a block, is created, added to the blockchain, and quickly published to all nodes. This allows bitcoin software to determine when a particular bitcoin amount has been spent, which is necessary in order to prevent double-spending in an environment without central oversight. Whereas a conventional ledger records the transfers of actual bills or promissory notes that exist apart from it, the blockchain is the only place that bitcoins can be said to exist in the form of unspent outputs of transactions.

What does mining mean?

The term “Mining “should instead of imagining a group of coal-darkened miners, instead be seen as a means of a record-keeping service. Miners keep the blockchain consistent, complete, and unalterable by repeatedly verifying and collecting newly broadcast transactions into a new group of transactions called a block. Each block contains a cryptographic hash of the previous block, using the SHA-256 hashing algorithm, which links it to the previous block, thus giving the blockchain its name.

In order to be accepted by the rest of the network, a new block must contain a so-called proof-of-work.
The proof-of-work requires miners to find a number called a nonce, such that when the block content is hashed along with the nonce, the result is numerically smaller than the network’s difficulty target.  This proof is easy for any node in the network to verify, but extremely time-consuming to generate, as for a secure cryptographic hash, miners must try many different nonce values (usually the sequence of tested values is 0, 1, 2, 3, …before meeting the difficulty target.

The proof-of-work system, alongside the chaining of blocks, makes modifications of the blockchain extremely hard, as an attacker must modify all subsequent blocks in order for the modifications of one block to be accepted. As new blocks are mined all the time, the difficulty of modifying a block increases as time passes and the number of subsequent blocks (also called confirmations of the given block) increases.

Now, what are Wallets? 

wallet stores the information necessary to transact bitcoins. While wallets are often described as a place to hold, or store bitcoins, due to the nature of the system, bitcoins are inseparable from the blockchain transaction ledger. A better way to describe a wallet is something that “stores the digital credentials for your bitcoin holdings, and allows one to access (and spend) them. Bitcoin uses public-key cryptography, in which two cryptographic keys, one public and one private, are generated. At its most basic, a wallet is a collection of these keys.

Besides software wallets, Internet services called online wallets offer similar functionality but may be easier to use. In this case, credentials to access funds are stored with the online wallet provider rather than on the user’s hardware. As a result, the user must have complete trust in the wallet provider.

Physical wallets store the credentials necessary to spend bitcoins offline. Examples combine a novelty coin with these credentials printed on metal. Others are simply paper printouts. Another type of wallet called a hardware wallet keeps credentials offline while facilitating transactions.

There are several different types of Bitcoin wallets, but the most important distinction is in relation to who is in control of the private keys required to spend the bitcoins. Some Bitcoin “wallets” actually act more like banks because they are holding the user’s private keys on their behalf.
If you choose to use one of these services, be aware that you are completely at their mercy regarding the security of your bitcoins. Most wallets, however, allow the user to be in charge of their own private keys. This means that no one in the entire world can access your account without your permission. It also means that no one can help you if you forget your password or otherwise lose access to your private keys. If you decide you want to own a lot of Bitcoin it would be a good idea to divide them among several different wallets. As they saying goes, don’t put all your eggs in one basket.