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1.
To do the 'Dirty Egyptian' is for one person (the masturbator) to simultaneously masturbate (often quite energetically) two other people using one hand on each recipient. The recipients are positioned behind and in front of the masturbator so when the deed is being carried out the instagator looks like they are walking like an Egyptian (ref the 1986 Bangles hit "Walk Like An Egyptian").

The Dirty Egyptian can sometimes be witnessed on the London underground late at night when passing through some of the less salubrious places on the network. This more public version is, however, normally carried out through clothing and generaly starts off as a frottage.
Chardonnay was being frottaged by a couple of guys on the tube last night so she decided to be a good sport and helped them to a dirty Egyptian.
by Orangenbaum March 08, 2013
 
2.
Basically, when you are blacked out in a bar, and a co-worker forces herself on you and jams her tounge down your throat.
"I was hanging at a bar on St.Pattys and some temp I know stumbled up and pulled a dirty egyptian! I was like, what the F."
by Mike G in da House April 03, 2008