In the context of this discussion, an askan is someone who gets involved in various individual or communal issues to try and resolve problems. Here are just a few examples:
Ploni was arrested for driving without his drivers license. He needs to be gotten out of jail, and allow the judicial processes to continue without incarceration.
There is a large affair, wedding, Hachnosas Sefer Torah, asifa, levaya r”l, which will require crowd control and assistance from various government and law enforcement agencies.
Some of the activities of known organizations such as Hatzoloh, Shomrim, etc. are askanus.
Issues in many of the frum communities are addressed by askanim (as opposed to officials of any sort), such as bike lanes in neighborhoods that are offended by such traffic, parking issues, communal locations for burning chometz, the famous islands that obstructed traffic near Maimonides Hospital, etc. These issues are generally brought to authorities, but it is tireless work of dedicated volunteers (a.k.a. askanim) that facilitate these solutions.