The word "handle" has at least two different (somewhat related) meanings in MATLAB.
A handle object is an object where different copies can refer to the same underlying "thing". So in Image Analyst's example, hFig is a handle object to the figure window on screen.
hFig = figure;
hFig2 = hFig;
Changing a property of hFig affects the (one) figure window on screen, and the property value of hFig2 reflects that change (and vice versa.)
before = hFig.Color
hFig2.Color = 'r';
after = hFig.Color
Even though it doesn't look like I did anything to hFig, I did do something to the underlying window on screen and so before and after are not the same.
See this documentation page for more information on how handle objects behave. Using handle ojects can be a bit of an advanced skill (using graphics objects to get and set properties of the underlying things on screen is a bit simpler, see this page for more information on how to do that) and writing your own handle classes is even more advanced.
A function handle is kind of sort of conceptually a handle object that refers to a function. I can use it to evaluate that function without hard-coding a call to the function into my code. Note that in the code below, just by changing fh and executing the exact same code I can do two very different things.
fh = @plot;
fh = @plus;
You can use the idea of a telephone number as an analogy for both of these meanings of handle. If I have your telephone number, and so does Image Analyst, that doesn't mean there are two copies of you running aroung the world. I could call you up and tell you something, then if Image Analyst were to call you you could tell them what I told you. Your telephone number is kind of a "handle to you".
Similarly, I'd get different behavior if I were to call you and say "I'd like to order a pizza for delivery" than if I called the local pizza place and give them the same message. Same data, different "function", different behavior.
For more information on function handles, see this documentation page. Often I think people's first or early experiences with function handles usually come with the ODE solvers, where the user enters a function handle as input so the solvers don't have a function hard-coded inside them.