What is a hokamuri?

I have seen this often in Japanese animation when a character ties a cloth under their nose before sneaking around. What is the purpose of it, what i it and where did it come from?

Basically a hokkamuri used to indicate that a character is a thief, or for comic effect. This site describes it well:

http://www.furinkan.com/kyokai/misc/references/hokkamuri.html

http://goo.gl/lU6DtG Google Image Search(lol

Yes, as yllwsmrf illustrated, tenugui((hand towel) for hokkamuri to hide face and a parcel wrapped in furoshiki on the back is common image of traditional thief in Japan.

By the way, also "hokkamuri" has meaning like "burying head in the sand" or "pretending not to see".

hokkamuri(ほっかむり),
hoo-kaburi(-kamuri)(ほおかぶり(-かむり)/頬被り/頬▽冠り)
are not means only "thief[泥棒/dorobou]" style.

hokkamuri have two meanings exist
1) calling "tenugui[てぬぐい/手拭い]" work style as general
2) " hokkamuri(hoo-kaburi)" as one of tenugui work style
http://dictionary.goo.ne.jp/leaf/jn2/202853/m0u/picture/0/
http://www.rock-de-mountain.jp/%E3%81%82%E3%82%8C%E3%81%93%E3%82%8C/
(tenugui : thin Japanese hand towel made of cotton.)

Thief style are calling "hana-kake(-gake)[鼻掛け]"
but most Japanese using/calling "hokkamuri" instead usual as mean 1).

This thief style have spreaded from "Nezumi Kozou" story (of Kabuki[歌舞伎] in Edo-period).
http://en.wikipedia.org/wiki/Nezumi_Koz%C5%8D
http://ja.ukiyo-e.org/image/waseda/007-1751 (Ukiyoe[浮世絵] in Edo-period)
http://www.shochiku.co.jp/cinemakabuki/lineup/01/ (present Kabuki poster)

TV anime "Roman" is this story based. (Lupin's gentleman spirit,too)
http://en.wikipedia.org/wiki/Bakumatsu_Gijinden_Roman
http://www.tv-tokyo.co.jp/anime/roman/index2.html

I've seen men wearing something similar while on parade in the Awa Odori. Is this the same item and is it called the same in that instance?

@John B

Yes, Awaodori Men's dance [男踊り/Otoko odori] using Hokkamuri(as hanakake) style a lot
but tenugui stlyle not only hanakake type so this is not "must" regulation(fes rules)
and no special name for this dance cos.
http://www.awanavi.jp/dancetraining.html

This team[ゑびす連/Ebisu(*1)-ren] women using,too. (not so rare case)
http://ebisu-awaodori.hacca.jp/?page_id=48
http://youtu.be/uWBseCQN4o4 (front vivid blue happi cos 3 young womens)

Hokkamuri(as hanakake) means only thief style just famous as Kabuki stage performance from.
Original hokkamuri (a lot of variations see last post link of images)
using for daily works(in field/kitchen/house) in old Japan
no western cap/hat/helmet culture period. Hanakake style is not regular use.
http://matome.hatena.ne.jp/mt/2641
http://en.wikipedia.org/wiki/Kasa_(hat) : Awa Women dance hat

And Japanese traditional hokkamuri using
in "Yasugi-bushi[安来節(*2)]" dance on present stage also very famous.
http://youtu.be/xAmpwyAiVWU
http://youtu.be/s1hUhefedhs
http://www.visit-matsue.com/nearby/yasugi.html
http://www.y-engeikan.com/
But this is not hanakake style.
This comedy dance means catching dojou[どじょう] pantomime with music.
http://en.wikipedia.org/wiki/Pond_loach
Yasugi-bushi means song title as minyo[民謡](old folk) Yasugi is land name in Shimane PF.
Dance name is "Dojo-sukui[どじょうすくい]" most adult Japanese knows well.

*1)
Ebisu[えびす/ゑびす/恵比寿] is one of seven-lucky-gods[七福神/Shichi-fuku-jin]
very very famous and custom in Japan culture.
http://en.wikipedia.org/wiki/Seven_Lucky_Gods
http://en.wikipedia.org/wiki/Ebisu_(mythology)
Japanese beer brand "YEBISU" and town Ebisu in Tokyo also famous same origin.
http://www.sapporobeer.jp/yebisu/
http://www.sapporobeer.jp/yebisu/history/

*2)
A lot of Japanese rememvered as "Yasu[k]i-bushi" but this is wrong
bc [来] reading [ki (as kun reading/訓読み:kun-yomi)] usual.