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) :
Method method = foo.getClass().getMethod("doSomething", null);
method.invoke(foo, null);
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.
There are some good reflection examples to get you started at http://docs.oracle.com/javase/tutorial/reflect/index.html
And finally, yes, the concepts are pretty much similar in other statically typed 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.
Update from a comment:
The ability to inspect the code in the system and see object types is
not reflection, but rather Type Introspection. Reflection is then the
ability to make modifications at runtime by making use of
introspection. The distinction is necessary here as some languages
support introspection, but do not support reflection. One such example
is C++
It sounds to me like your web host is not very flexible. I would ask them about adding the needed permissions they allow. I had the same problem with Server Intellect and after speaking with them and telling them the permission I needed, they helped out and worked it out.
Honestly though, there are a lot of hosting companies and a lot who are willing to work with people to get their business. If I were facing the same difficult hosting provider, I would look elsewhere to a provider that will help you out. The change they made caused you pain and they should at least attempt to help you, otherwise move on to someone who wants to support you. Just me 2 cents, probably not a recommendation you were looking for.
Best Answer
Reflection allows malicious code to inspect all kinds of secrets: not so much intellectual property (though sure, that too), but data that should be private and secure, like connection strings, passwords, bank account data, etc..
Of course, many programs expose this data as a matter of course through even more-easily compromised vectors, but there's no reason to increase an application's attack surface.
Edited to bring some of the conversation up from the comments:
It's probably true that the real risk is unrestricted file system access, which is what turns reflection into a real danger. If a bad actor can get an assembly (or something that gets compiled into an assembly) into your virtual directory, you're in trouble if they have reflection permission. (Of course if this happens, there are other potential problems as well, but that shouldn't discount this particular vulnerability.)
In a shared hosting environment that's just harder to prevent, though it certainly isn't impossible. Perhaps it's worth cross-posting this question to ServerFault to see what the good folks there have to say.