The web application I am building has a custom right-click menu. I'm concerned that the user might not find this menu as normally right-clicking will only display the default browser's menu. Is there a standard way to make the user aware of custom right-click menus? Is there another way to show this information?
If you're worried people won't discover the context menu action, how about having a small button in the corner of each actionable item when you hover over it. This would pop up the context menu.
Context menus are an added benefit for advanced users - this is even the case on desktop applications (and even the OS itself). Even fairly experienced users of Windows tend to ignore context menus almost entirely, so I wouldn't worry too much about the fact that a web browser has a "standard" context menu that you would be overriding. If the current user understands context menus and knows how to use them, they will have no problem understanding that the context is now different, which is why they see different options (hence the name "context menu").
An important caveat is that you should never use context menus as the exclusive means to complete a task, but adding them provides your advanced users with an easier and quicker way to get things done.
There is absolutely no problem with using a context menu, and if it makes sense for your application (web or desktop) then go ahead and use it. The key thing is to make sure that every action in your context menu can be completed in another manner for those users that don't really "get" context menus (or for whatever reason don't use them).
Depending on the application, I would even go a step further and add keyboard shortcuts for certain commands as well. If you're getting to the point of setting up context menus, you are probably at the point where you are optimizing your interface, so keyboard shortcuts can only help with that more.
Generally I would suggest you not use right click menus, because just like you said: users are not aware of this menu's existence.
Jakob's Law of the Web User Experience states that "users spend most of their time on other websites." This is the biggest reason for you to use another solution
On the other hand, if it's a corporate app or an app where users need to study the UI anyway, or if the UI is like Excel or any other Office apps, you may try using a unique right click menu because in this case users would expect your app to function like the offline ones.
It depends on the nature of your web app, and the true answer can only be bore out with proper user testing.
It should be fine if your app is an 'editor tool': If your app is a tool of some kind that has a 'canvas' or 'grid' like a document editor, spreadsheet, or drawing tool, then I suspect you'll find users will naturally expect the right click to work and be context-sensitive to your specific app's needs. That the browser menu drops down outside of the canvas area is fairly intuitive.
Well established existing precedents exist: E.g. most if not all of Google's online 'editor tools' (docs, spreadsheets, drawings etc) support right click.
PS labelling niggle: it's not a right click for left-handed users. Sadly, for the more sinister amongst us it is not a right click, but they are used to it being called this so I guess it's unavoidable really.