Let's say you go to the dentist to have a tooth pulled out.
When the receptionist asks you for your name, that's information they need in order to begin the appointment. In this example, your name is contextual information. So in the context of visiting the dentist, you need to provide your name to get your tooth pulled.
Now let's say you walk over to the bank.
At the bank, you ask to withdraw $100. The teller needs to establish your identity before giving you money, so you'll probably have to show them a driver's license or swipe your ATM card and enter your PIN number. Either way, what you're providing is context. The teller uses this information to move the transaction forward. They may then ask you which account you'd like to withdraw from. When you answer, "My savings account", that's even more context.
The more context you give, the more knowledge the other party has to help deal with your request. Sometimes context is optional (like typing more and more words into your Google search to get better results) and sometimes it's required (like providing your PIN number at the ATM). Either way, it's information that usually helps to get stuff done.
Now let's say you take your $100 and buy a plane ticket to fly somewhere warm while your mouth heals.
You arrive at a nice sunny destination, but your bag doesn't make it. It's lost somewhere in the airport system. So, you take your "baggage claim ticket" (that sticker with the barcode on it) to the "Lost Baggage office". The first thing the person behind the desk will ask for is that ticket with your baggage number on it. That's an example of some required context.
But then the baggage person asks you for more information about your bag like so they can find it more easily. They ask, "What color is it? What size is it? Does it have wheels? Is it hard or soft? While they don't necessarily need those pieces of information, it helps narrow things down if you provide them. It reduces the problem area. It makes the search much faster. That's optional context.
Here's the interesting part: for a lot of software and APIs, the required context usually ends up as actual parameters in a method signature, and optional context goes somewhere else, like a flexible key-value map that can contain anything (and may be empty) or into thread-local storage where it can be accessed if needed.
The examples above are from real life, but you can easily map them to areas within computer science. For example, HTTP headers contain contextual information. Each header relates to information about the request being made. Or when you're sending along a global transaction ID as part of a two-phase commit process, that transaction ID is context. It helps the transaction manager coordinate the work because it's information about the overall task at hand.
Hope that helps.