What are unusual items?
As nimcap answered, unusual hats are just hats with special particle effects. They have no stat benefits, only visual effects.
The only way to obtain them is to trade for them or to find one in a Mann Co Supply Crate. Crates have a low chance to contain an unusual hat.
According to data gathered by users on the forums, it seems that series 1 crates have about 2.45% chance of containing an unusual item, and series 2 crates have about a 1.5% chance. You can tell which series of crate you have by reading the tooltip.
To open the crate you need to buy a Mann Co. Supply Crate Key, which costs $2.49 in the store. If you'd rather not spend money you can trade the crate to someone. Items that come out of crates are craftable and tradeable.
You can find more details about the drop rate on the TF2 wiki page about crates. Keep in mind that this data is gathered by players and might not be totally accurate, but it's probably in the ballpark.
I'm actually copying and pasting part of my answer to yesterday's Vintage items question here...
Unusual hats were added to the game during the Mann-conomy update on September 30th.
These are hats have particle effects attached to them, such as ghosts or a sunbeam. The names for these items are shown in purple. They are not available through the random drop system, but instead as a rare drop from a Mann Co. Supply Crate, which require you to buy keys from the Mann Co. Store (for $2.49 each) to open. Consult the list below to know which effects are dropped from which crates.
Of course, like all hats, they can be traded. Expect to trade a lot of keys/earbuds for one, though, as they are exceedingly rare.
If you are trading to try and get Unusual hats, note that items can now be renamed... check the name color (should be purple for Unusual items) to make sure it really is an Unusual hat.
Also, there have been scammers out there lately who will put in an item and then change it while you are putting your items into the trade window. Always double-check what items are in the other person's side of the trade window before accepting.
The list of available Unusual hat effects are:
- Series #1 (Crates #1-25)
- Haunted Ghosts
- Circling TF Logo
- Circling Heart
- Circling Peace Sign
- Sunbeam
- Massed Flies
- Green Confetti
- Purple Confetti
- Green Energy
- Purple Energy
- Burning Flames
- Scorching Flames
- Searing Plasma
- Vivid Plasma
- Series #2 (Crates #26-57)
- Stormy Storm
- Blizzardy Storm
- Bubbling
- Smoking
- Steaming
- Nuts n' Bolts
- Orbiting Planets
- Orbiting Fire
- Series #3 (Crates #59-91 (except #87))
- Aces High
- Disco Beatdown
- Miami Nights
- Dead Presidents
- Cloud 9
- Kill-a-Watt
- Terror-Watt
- Robo (Crate #58 - Robo Community Crate)
- Anti-Freeze
- Roboactive
- Memory Leak
- Phosphorous
- Sulphurous
- Overclocked
- Power Surge
- Electrostatic
- Green Black Hole
- Time Warp
- End of the Line (Crate #87 - End of the Line Community Crate)
- Death At Dusk
- Morning Glory
- Frostbite
- Molten Mallard
- Taunts (Mann Co. Audition Reel, Mann Co. Director's Cut Reel, End of the Line Community Crate)
- '72
- Fountain of Delight
- Holy Grail
- Mega Strike
- Midnight Whirlwind
- Silver Cyclone
- Screaming Tiger
- Skill Gotten Gains
- Showstopper
- Halloween (All normal crates during the Halloween Event) Number in parentheses is the year the effect was added.
- Cauldron Bubbles (2011)
- Cloudy Moon (2011)
- Eerie Orbiting Fire (2011)
- Flaming Lantern (2011)
- Harvest Moon (2012)
- It's a Secret To Everybody (2012)
- Knifestorm (2012)
- Misty Skull (2012)
- Stormy 13th Hour (2012)
- Arcana (2013)
- Chiroptera Venenata (2013)
- Something Burning This Way Comes (2013)
- Poisoned Shadows (2013)
- Darkblaze (2013)
- Demonflame (2013)
- Hellfire (2013)
- Spellbound (2013)
- Amaranthine (2014)
- Bonzo The All-Gnawing (2014)
- Haunted Phantasm Jr. (2014)
- Ghastly Ghosts Jr. (2014)
- Stare From Beyond (2014)
- The Ooze (2014)
- Halloween Taunts (All taunt-dropping crates during the Halloween Event)
- Ghastly Ghosts
- Haunted Phantasm