Java Reflection makes it possible to inspect classes, interfaces, fields and methods at runtime, without knowing the names of the classes, methods etc. at compile time. It is also possible to instantiate new objects, invoke methods and get/set field values using reflection.
The name reflection is used to describe code which is able to inspect other code in the same system (or itself).
For example, say you have an object of an unknown type in Java, and you would like to call a 'doSomething' method on it if one exists. Java's static typing system isn't really designed to support this unless the object conforms to a known interface, but using reflection, your code can look at the object and find out if it has a method called 'doSomething' and then call it if you want to.
So, to give you a code example of this in Java (imagine the object in question is foo) :
One very common use case in Java is the usage with annotations. JUnit 4, for example, will use reflection to look through your classes for methods tagged with the @Test annotation, and will then call them when running the unit test.
And finally, yes, the concepts are pretty much similar in other statically types languages which support reflection (like C#). In dynamically typed languages, the use case described above is less necessary (since the compiler will allow any method to be called on any object, failing at runtime if it does not exist), but the second case of looking for methods which are marked or work in a certain way is still common.