I tried a few pair of $25-50 earphones that were marketed as sport or sweat resistant. But they crapped out after 6-10 months like all my other headphones. So now I went super cheap and found some Kos brand on sale for $6 on Amazon, and bought 5 pair. So far, the first pair is still going strong at 10 months.

Get something that is on the outside of your ears. They are cheap and safe.

I have the MEElectonics ones linked above, and I love them. They come with six sets of ear pieces, so you can choose ones that fit your ears, and a small carrying case. They also have a pretty decent microphone on them.

I like these - great for running.

http://www.amazon.com/Sony-MDR-AS20J-Active-Headphones-Hangers/dp/B0015AHRFK/ref=sr_1_11?ie=UTF8&qid=1400819166&sr=8-11&keywords=sony+over+the+ear+headphones

Yurbuds in ear headphones. These are great quality for the price and don't budge when running: http://www.amazon.com/gp/aw/s/ref=is_s_ss_i_0_3?k=yurbuds&sprefix=yur

I really like these, but they are slightly over $25 on Amazon:
http://www.amazon.com/Womens-Yurbuds-Inspire-Sport-Earphones/dp/B00AAEUEQQ/ref=sr_1_2?ie=UTF8&qid=1400589450&sr=8-2&keywords=yurbuds

They are small enough to stay in my tiny ears. They are even comfortable under an earband in the winter. I use them to listen to mostly pop music at pretty low volume (music is easily drowned out by buses and airplanes) while running. The sound quality probably does not compare to Bose, but is fine for my purposes.

these are good cheap headphones

http://www.amazon.com/Panasonic-RPHJE120K-In-Ear-Headphone-Black/dp/B003EM8008/ref=pd_bxgy_e_text_y

or if you want something labeled for running, maybe these

http://www.amazon.com/MEElectronics-Sport-Fi-Noise-Isolating-Headphones-Memory/dp/B0038W0K2K/ref=sr_1_1?s=aht&ie=UTF8&qid=1400588889&sr=1-1

I'm a casual runner, but need something better than my iPhone earbuds that keep coming out. Any suggestions for something inexpensive, but does the job of playing decent quality, and can handle sweat?