CSS selector for an element having class .a and class .b

div[class~="a"][class~="b"] {
    color: #f00;
}

Class selectors can be combined:

div.a.b {
  color: red;
}

Quoting from the spec:

To match a subset of "class" values, each value must be preceded by a ".".

For example, the following rule matches any P element whose "class" attribute has been assigned a list of space-separated values that includes "pastoral" and "marine":

p.marine.pastoral { color: green }

This rule matches when class="pastoral blue aqua marine" but does not match for class="pastoral blue".

That's entirely possible. If you specify two classes on an element (without any spaces), that means that it must have both for the rule to apply.

div.a {
  color: blue;
}
div.b {
  color: green;
}
div.a.b {
  color: red;
}

<div class="a">
  A
</div>
<div class="b">
  B
</div>
<div class="a b">
  AB
</div>