We use cryptographic techniques daily without really knowing how they work, so I'm going to try and explain some basic concepts. Let's start with Wikipedia's current definition:
Cryptography or cryptology is the practice and study of techniques for secure communication in the presence of third parties called adversaries.
One cryptographic process we are all familiar with is encryption, that allows us to change the contents of a message so only certain people with a "key" can decipher and read it. A simple—and well known—example of encryption is the Caesar cipher (if you haven't heard of it, check out how it works!).
Let's consider the following scenario with three people (or parties): Alice, Bob and Craig. Alice wants to contact Bob privately, while Craig is trying to eavesdrop. This is all happening through a network, in this particular scenario, they are communicating through the mail. Craig works at the postal office, so he could get Alice's letter, open it, read it, put it back in a new envelope that looks exactly the same as Alice's and then send it to Bob.
Craig's attack is known as a man-in-the-middle attack, happening when the attacker is able to secretly relay information between two parties (and with the ability to change the contents of the communication). This attack isn't particularly hard to carry out on the Internet, but we are normally protected by cryptographic methods (that ensure the privacy and authenticity of our communications).
Alice knows about the flaws of the mail system, so she decides to encrypt her message. She could use the Caesar cipher. If Bob knows how much Alice "shifted" the alphabet, he will be able to read her message, while Craig won't. Or will he? Couldn't Craig just try all the numbers from 1 to 25 and just see which one gives a message that makes sense? And how did Alice tell Bob how much she "shifted" the alphabet without Craig reading it?
Those are good points. We currently use better encryption methods than the Caesar cipher that tackle these issues. The first concern is talking about a brute-force attack (when the attacker tries many keys in order to—eventually—find the correct one). We can protect our messages against brute-force attacks by using an encryption method that admits a huge number of different possible keys. How big? If you create a key with GPG, the minimum key size is 1024 bits (which gives us 21024 different possible keys). How hard would it be to crack it? This video explains it pretty well for a key that is 256 bits long (2256 possible keys). First problem solved! Bob isn't deciphering our letter anytime soon!
About the second issue... How can Alice tell Bob her secret password before they can encrypt anything? It turns out she doesn't need to do that at all! She can use asymmetric cryptography to solve this problem. In asymmetric encryption, everyone has two keys1: a public key and a private key. Our public key will be public! Everyone can know it (and that won't put our encrypted messages in danger), while our private key will only be known to us. When using asymmetric encryption, we encrypt messages using someone else's public key, but only someone with the private key will be able to decipher it.
So now Bob can simply send Alice his public key, which she will use to encrypt the message. Only Bob with his private key will be able to decipher the message. A system of communication that is resistant to Craig's attacks, so far...
Craig can't decipher the message, so he might try another strategy: change it! He will get Alice's letter, destroy it, and send a different one to Bob (making it look like it came from Alice). The communication is private, but not secure yet!
Once again, cryptographic techniques come to the rescue with the ability to digitally sign messages (also using asymmetric cryptography). What signing a message does is kind of the opposite of encryption: Alice can use her private key to sign her message, which will output a new file (the signature). Now, anybody with the message, the signature made by Alice, and her public key can check that the message was signed using Alice's private key, therefore ensuring nobody changed it (signatures are different for different messages).
Now, Craig can still destroy the message and send a different one. However, Bob will realize there isn't a signature (or the one given doesn't match the message). This will alert Bob that the contents of the message might indeed not come from Alice. Bob might not be able to get Alice's message, but Craig will never be able to impersonate her.
The problem with the digital signature is that there has to be an initial contact that both parties know has not been compromised2. This could be achieved by meeting in person and exchanging keys, although that could be hard for two parties that live in different parts of the world trying to talk over the Internet. There are methods to work around this problem, although none is perfect.
Hopefully, this post gave you a basic overview of some things that can be done using cryptographic techniques and how they are necessary when securing our online communications.
Edit: Invidious link has been changed to YouTube as Invidious instance is shutting down.