strongly vs dynamically typed
This is often misunderstood so let me clear it up.
Static/Dynamic Typing
Static typing is where the type is bound to the variable. Types are checked at compile time.
Dynamic typing is where the type is bound to the value. Types are checked at run time.
So in Java for example:
String s = "abcd";
s will "forever" be a String. During its life it may point to different Strings (since s is a reference in Java). It may have a null value but it will never refer to an Integer or a List. That's static typing.
In PHP:
$s = "abcd"; // $s is a string
$s = 123; // $s is now an integer
$s = array(1, 2, 3); // $s is now an array
$s = new DOMDocument; // $s is an instance of the DOMDocument class
That's dynamic typing.
Strong/Weak Typing
(Edit alert!)
Strong typing is a phrase with no widely agreed upon meaning. Most programmers who use this term to mean something other than static typing use it to imply that there is a type discipline that is enforced by the compiler. For example, CLU has a strong type system that does not allow client code to create a value of abstract type except by using the constructors provided by the type. C has a somewhat strong type system, but it can be "subverted" to a degree because a program can always cast a value of one pointer type to a value of another pointer type. So for example, in C you can take a value returned by malloc() and cheerfully cast it to FILE*, and the compiler won't try to stop you—or even warn you that you are doing anything dodgy.
(The original answer said something about a value "not changing type at run time". I have known many language designers and compiler writers and have not known one that talked about values changing type at run time, except possibly some very advanced research in type systems, where this is known as the "strong update problem".)
Weak typing implies that the compiler does not enforce a typing discpline, or perhaps that enforcement can easily be subverted.
The original of this answer conflated weak typing with implicit conversion (sometimes also called "implicit promotion"). For example, in Java:
String s = "abc" + 123; // "abc123";
This is code is an example of implicit promotion: 123 is implicitly converted to a string before being concatenated with "abc". It can be argued the Java compiler rewrites that code as:
String s = "abc" + new Integer(123).toString();
Consider a classic PHP "starts with" problem:
if (strpos('abcdef', 'abc') == false) {
// not found
}
The error here is that strpos() returns the index of the match, being 0. 0 is coerced into boolean false and thus the condition is actually true. The solution is to use === instead of == to avoid implicit conversion.
This example illustrates how a combination of implicit conversion and dynamic typing can lead programmers astray.
Compare that to Ruby:
val = "abc" + 123
which is a runtime error because in Ruby the object 123 is not implicitly converted just because it happens to be passed to a + method. In Ruby the programmer must make the conversion explicit:
val = "abc" + 123.to_s
Comparing PHP and Ruby is a good illustration here. Both are dynamically typed languages but PHP has lots of implicit conversions and Ruby (perhaps surprisingly if you're unfamiliar with it) doesn't.
Static/Dynamic vs Strong/Weak
The point here is that the static/dynamic axis is independent of the strong/weak axis. People confuse them probably in part because strong vs weak typing is not only less clearly defined, there is no real consensus on exactly what is meant by strong and weak. For this reason strong/weak typing is far more of a shade of grey rather than black or white.
So to answer your question: another way to look at this that's mostly correct is to say that static typing is compile-time type safety and strong typing is runtime type safety.
The reason for this is that variables in a statically typed language have a type that must be declared and can be checked at compile time. A strongly-typed language has values that have a type at run time, and it's difficult for the programmer to subvert the type system without a dynamic check.
But it's important to understand that a language can be Static/Strong, Static/Weak, Dynamic/Strong or Dynamic/Weak.