If you fill a glass with ice cubes (or even ground ice), all the way to the top, and then pour a little water in, it doesn't overflow. This is because of all of the interstices between the bits of ice, which are full of air (unless you're in a vacuum or something).

This writeup pertains partly to the interstitial space in a crystalline lattice. Ionically bonded materials are full of interstices. Since they are made of approximately spherical objects, as the spheres pack together, there is plenty of room in between.

These interstitial spaces have varying sizes, depending on the structure of the lattice, and the size(s) of the atoms of the element(s) in the lattice. The two main types of interstitial space in crystalline lattices are tetrahedral and octahedral.

A tetrahedral interstice has 4 nearest neigbors, forming the 4 vertices of the tetrahedron. An octahedral interstice has 6 nearest neighbors. This might seem counterintuitive, but in fact only 6 vertices are needed to make an 8-sided shape in three dimensions. Think of it as a square, laying flat in the plane of your monitor.

+-----------+
|               |
|               |
|      +<- - - - - - sticking out of the monitor
|               |
|               |
+-----------+

This, if flipped over completely would have the current out-sticking atom sticking into the monitor, and another one sticking out. Here's a side-on view:

+
/|\
/ | \
/  |  \
/   |   \
**+    |    +**
\   |   /
\  |  /
\ | /
\|/
+

Where the atoms with the stars on them are the ones previously sticking into and out of the monitor. The second diagram's a little stretched due to the nature of / and \ characters.

One of the really important things about this interstitial space is that interstitial impurity atoms can fit inside of them, effectively dissolving into the solid lattice. The shape and size will help determine what can fit inside them.

Derived from my Materials Science and Engineering lecture notes.