I can’t speak to Nebraska’s system of nonpartisan elections. But I chair one of the major political parties in Minneapolis, where the system is probably similar, since both states come from a common political tradition whose roots lie in the Grange movement of the late 19th century and the Progressive movement of the early 20th century. The Minneapolis City Council and other municipal offices are all elected on a nonpartisan basis, as was the Minnesota Legislature until the early 1970s.
“Nonpartisan” elections in Minnesota doesn’t mean that the election occurs without partisan political involvement. Far from it: each party endorses candidates for office, and these endorsements and the candidates’ partisan affiliation are generally known to the voters. An election is nonpartisan, though, in the sense that candidates are nominated and elected without regard to political party. Thus, where a primary election narrows a multicandidate field down to two, both candidates who advance to the general electon can come from the same political party – as opposed to a partisan system, where the primary serves the purpose of narrowing the field to one candidate per party, and the primary is effectively how each party nominates its official standard-bearer. (I believe that Louisiana uses a similar system for some elections, although I don’t know the details.)
A candidate can communicate his or her partisan affiliation in various ways, formal and informal. The law provides for some nonpartisan elections in Minnesota that a candidate may list a “political party” or a “political principle” of up to three words on the ballot; many candidates use this opportunity to list their political party. Some political parties publish, and circulate as widely as they can afford, a “sample ballot” listing their endorsed candidates.