Java is a statically typed language, so the compiler does most of this checking for you. Once you declare a variable to be a certain type, the compiler will ensure that it is only ever assigned values of that type (or values that are sub-types of that type).
The examples you gave (int, array, double) these are all primitives, and there are no sub-types of them. Thus, if you declare a variable to be an int:
int x;
You can be sure it will only ever hold int values.
If you declared a variable to be a List, however, it is possible that the variable will hold sub-types of List. Examples of these include ArrayList, LinkedList, etc.
If you did have a List variable, and you needed to know if it was an ArrayList, you could do the following:
List y;
...
if (y instanceof ArrayList) {
...its and ArrayList...
}
However, if you find yourself thinking you need to do that, you may want to rethink your approach. In most cases, if you follow object-oriented principles, you will not need to do this. There are, of course, exceptions to every rule, though.