I was browsing through the wardrobe, and I saw a bunch of avatar outfits that had a "monthly avatar item" icon in the corner. What does this mean and how do I get them? Also, what is this month's item?

Basically, the site releases a set of items (usually some sort of theme) every month or so. They're only available for purchase that month (and possibly in the Black Friday sale towards the end of the year).
The Site Administrators will often make an announcement about each months items: example 1, example 2, example 3. All users are pinged for an announcement so you should know when the next one is made.

The monthly items are available for diamonds in the Avatar Shop.

There isn't a set for this month.