Monero is a secure, private, and untraceable cryptocurrency. It is open-source and accessible to all. With Monero, you are your own bank. Only you control and are responsible for your funds. Your accounts and transactions are kept private from prying eyes.
Monero (XMR) is a Cryptonote algorithm based cryptocurrency, it relies on Ring Signatures in order to provide a certain degree of privacy when making a transaction. Monero is a Proof of Work cryptocurrency that can be miner with computational power from a CPU or GPU. There are currently no ASICs for Monero, which means that anyone with a computer can mine it.
The first thing you need to think about is if you are going to mine solo or on a pool. Mining on a pool usually comes with a fee but also has it's advantages. Users that don't have enough computational power may have to wait long periods of time before getting a reward, in a pool users pull their efforts together in order to reduce variance and have a steady flow of income. Users that have enough hardware or simply want to mine solo can also benefit from solo pools which function in the same way as normal solo mining, but don't require users to run a full node or store the entire blockchain. We are going to use MoneroHash for this guide, but you can use any pool you like.
Monero is secure.
Monero is a decentralized cryptocurrency, meaning it is secure digital cash operated by a network of users. Transactions are confirmed by distributed consensus, and then immutably recorded on the blockchain. Third-parties do not need to be trusted to keep your Monero safe.
Monero is private.
Monero uses ring signatures and ring confidential transactions to obfuscate the amounts, origins, and destinations of all transactions. Monero provides all the benefits of a decentralized cryptocurrency, without any of the typical privacy concessions.
Monero is untraceable.
Sending and receiving addresses as well as transacted amounts are obfuscated by default. Transactions on the Monero blockchain cannot be linked to a particular user or real-world identity.
Monero is fungible.
Monero is fungible because it is private by default. In its current state, it is extremely unlikely that Monero will ever be blacklisted by exchanges or vendors due to its association in previous transactions.
In cryptography, a ring signature is a type of digital signature that can be performed by any member of a group of users that each have keys. Therefore, a message signed with a ring signature is endorsed by someone in a particular group of people. One of the security properties of a ring signature is that it should be computationally infeasible to determine which of the group members' keys was used to produce the signature.
For instance, a ring signature could be used to provide an anonymous signature from "a high-ranking White House official", without revealing which official signed the message. Ring signatures are right for this application because the anonymity of a ring signature cannot be revoked, and because the group for a ring signature can be improvised (requires no prior setup).
Application to Monero
A ring signature makes use of your account keys and a number of public keys (also known as outputs) pulled from the blockchain using a triangular distribution method. Over the course of time, past outputs could be used multiple times to form possible signer participants. In a "ring" of possible signers, all ring members are equal and valid. There is no way an outside observer can tell which of the possible signers in a signature group belongs to your account. So, ring signatures ensure that transaction outputs are untraceable. Moreover, there are no fungibility issues with Monero given that every transaction output has plausible deniability (e.g. the network can not tell which outputs are spent or unspent).
To read how Monero gives you privacy by default (unlinkability), see stealth addresses.