I tend to build my house nearly completely out of glass because I like to be able to see all nearby threats. Are there any monsters with the ability to break glass? If so, which ones?
Glass is a fragile block when compared to other ones. Looking at this page on the Minecraft Wiki which lists every block and their resistances to explosions, Glass blocks have a blast resistance rating of 1.5. There are two mobs in the game which either explode or shoot projectiles that explode: the Creeper and the Ghast. In order to not be broken by a Ghast's fireballs, a block must have a resistance rating of 20.17 or higher, and a Creeper's explosion is even more powerful than that. Because of this, you have to be careful to not let Creepers explode near or Ghasts shoot at your glass structures.
The other two destructive mob in Minecraft are the Enderman and Silverfish. Endermen can pick up certain blocks and place them down elsewhere, but Glass is not one of the blocks they can pick up. Silverfish can hide in Stone, Cobblestone, and Stone Brick blocks which get destroyed when they appear from them, so they have no effect on Glass.
As an additional note to Kevin's response, while ghasts can break glass, according to the wiki they don't have much reason to:
Ghasts will not fire at the player without a line-of-sight - this can be blocked by glass, glass panes, leaves, Nether portals, vines, iron bars, both types of fences, and cobwebs - making safe scenic paths through the Nether a viable possibility.
equally, this page http://www.minecraftwiki.net/wiki/Creeper has a lovely discussion towards the bottom about creepers and buildings. The salient details? glad you asked!
Since wooden slabs are a damage value of stone, They can be used to make wooden creeper-resistant houses.This no longer works as of 12w17a.
Glass walls around entrances can preempt the creepers' habit of hiding in ambush around corners (but this is offset by the disadvantage that glass is not very durable in an explosion).
Placing a small glass wall 2 or 3 blocks high around the area wanted to be protected can be a very effective method. Just don't get within the area where your wall is and keep the area being enclosed well lit.
(the above is because, apparently, creepers only explode when they're near you, not when they're blocked by walls)
Fencing provides a great way to protect holdings, provided that the player finds fencing aesthetically compatible with a given property.
A ring of cats around a fortress will keep creepers at a safe distance at all times. While somewhat time-intensive, this is the most effective way at dealing with creepers, although other monsters can still draw close to your base at night.
I had this same question, so I did a test. I made a glass box, and stayed in it over night. I saw a zombie, who didn't seem to notice me, so I broke the glass and walked over to get his attention, then ran back into the glass box and put new glass in the holes I had made to seal myself in. The zombie then acted as if he couldnt even see me.
A few minutes later I saw a creeper and ran out to get his attention, then ran back and sealed myself in the box at the last second, and he immediately de-aggroed me and simply walked away.
I did this all on Hard difficulty survival mode. I wanted to know the question, so I could build a box to be safe in and AFK while my crops grew in my huge farm tower!!