The issue is somewhat complicated.

If Plot is given something that isn't the correct form, it makes a blank plot.

The rest is somewhat difficult to explain without more experience. Your code above doesn't put variables into a proper context. Contexts are much like namespaces in other languages. In the example below, I have "MathExpresion" put all the variables into a context called "LocalContext" and then use that context in your code:

CloudDeploy[
 FormFunction[{"equation" -> Restricted["MathExpression", Automatic, Automatic, "LocalContent`"]}, 
             Plot[#equation, {LocalContent`x, -10, 10}] &, 
             "PNG"]
]

The & operator defines something called a pure or anonymous function.

Is there somewhere you copied this code from? I'd like to check that it explains some of this if possible.

Restricted depends heavily on the type (its first argument). The documentation for Restricted doesn't cover its full behavior, just a very large set of common ways that it is used.

Look at the documentation for things like "Expression" and "MathExpression" under "Restriction Specifications". These documentation pages cover how Restricted is used with them.

http://reference.wolfram.com/language/ref/interpreter/Expression.html

The documentation pages for Interpreter types also have examples of how Restricted is used with them.

I would start by making graphs and similar things. CloudDeploy and FormFunction are useful once you've learned the basics of the language.

I would take a look at the virtual book for an introduction:

http://reference.wolfram.com/language/tutorial/VirtualBookOverview.html

There are more resources in books and videos you might find helpful:

https://www.wolfram.com/broadcast/screencasts/handsonstart/