Differentiable
Differentiable means that the derivative exists ...
Example: is x^{2} + 6x differentiable?
Its derivative is 2x + 6
(Because Derivative rules tell us the derivative of x^{2} is 2x and the derivative of x is 1.)
So yes! x^{2} + 6x is differentiable.
... and it must exist for every value in the function's domain.
DomainIn its simplest form the domain is 
Example (continued)
When not stated we assume that the domain is the Real Numbers.
For x^{2} + 6x, its derivative of 2x + 6 exists for all Real Numbers.
So we are still safe ... x^{2} + 6x is differentiable.
But what about this:
Example: The function f(x) = x (absolute value):
x looks like this: 
At x=0 it has a very pointy change!
Does the derivative exist at x=0?
Testing
We can test any value "c" by finding if the limit exists:
lim h→0 
f(c+h) − f(c) 
h 
Example (continued)
Let's calculate the limit for x at the value 0:
lim h→0 
0+h − 0  =  lim h→0 
h − 0  =  lim h→0 
h 
h  h  h 
To see why, let's compare left and right side limits:
From Left Side:  From Right Side:  


The limits are different on either side, so the limit does not exist.
So the function f(x) = x is not differentiable
A good way to picture this in your mind is to think:
As I zoom in, does the function tend to become a straight line?
The absolute value function stays pointy even when zoomed in.
Other Reasons
Here are a few more examples:
The Floor and Ceiling Functions are not differentiable at integer values, as there is a discontinuity at each jump. But they are differentiable elsewhere. 
The Cube root function x^{(1/3)} Its derivative is (1/3)x^{(2/3)} (by the Power Rule) At x=0 the derivative is undefined, so x^{(1/3)} is not differentiable. 
At x=0 the function is not defined so it makes no sense to ask if they are differentiable there. To be differentiable at a certain point, the function must first of all be defined there! 
As we head towards x = 0 the function moves up and down faster and faster, so we cannot find a value it is "heading towards". So it is not differentiable. 
Different Domain
But we can change the domain!
Example: The function g(x) = x with Domain (0,+∞)
The domain is from but not including 0 onwards (all positive values).
Which IS differentiable.
(And I am "absolutely positive" about that.)
So the function g(x) = x with Domain (0,+∞) is differentiable.
We could also restrict the domain in other ways to avoid x=0 (such as all negative Real Numbers, all nonzero Real Numbers, etc).
Why Bother?
Because when a function is differentiable we can use all the power of calculus when working with it.
Continuous
When a function is differentiable it is also continuous.
Differentiable ⇒ Continuous
But a function can be continuous but not differentiable. For example the absolute value function is actually continuous (though not differentiable) at x=0.