Having grown up "heimish" I will do my best to explain.
The first thing I tell people that ask me to define Heimish, is "mixed up". From the outside looking in, our accent in davening is typically that of chassidim, yet we (For the most part) are clean shaven (which is a huge no-no in the chassidish world). You might see us wear a gartel on shabbos (chassidish) while wearing a suit and tie (not-chassidish). Heimish people also frequent going to rebbes. So, yes we're sort of mixed up.
From my experience, there are really two kinds of backgrounds to Heimish ppl.
1) After the war, many people who were brought up full-on chassidish, when they moved to various parts of Europe, or came to the states, they either weren't comfortable wearing the chassidish get-up or whatever their reasons were, they decided to shave and wear the more litvish style of clothing. These people, while all their minhagim and traditions were all Chassidish, they didn't dress the part anymore. However, the chassidish way of life was really all they knew.
2) Back in Hungary, there were many towns and shtetls that bordered around the bigger chassidish towns and these people all gravitated towards the rebbes in the larger towns, but they themselves weren't full on chassidish. Litvish however they were not, as all the customs they followed were from the chassidim around them.
In our generation though, the whole concept of Heimish is really dying out. Growing up, there were many yeshivas that were chassidish yeshivas, where the rebbeim spoke Yiddish, and half the class was actually chassidish (think of Yagdil Torah, Veen etc) however the other half of the class had payis behind the ears (litvish style) and were considered "Heimish". What happened over the years though was that most started to feel this "mixed up" feeling and sort of chose a path. Many went to litvish yeshivas for high school/beis midrash and started acceping the litvish way of life, while the other half went to more chassidish yeshivas, and started growing a beard and put on a shtreimel when they got married. Our kids generation therefore is pretty well defined. There are very few if any (Veretsky in brooklyn being the exception) yeshivas that have chassidish yiddish speaking rebbeim, with the parent body being clean shaven (Montreal still has a little bit of that as well, however in Brooklyn/Lakewood etc it's pretty much over).
For me personally, to this day I feel mixed up. I speak a fluent Yiddish, have a chassidish havara, yet I live in Lakewood in a total litvish environment. I don't fit into bp anymore as that became very Chassidish (I'm clean shaven and not chassidish at all) but I feel like an outsider here in Lakewood as well since at my core, all my customs, my whole upbrining was very different than that of the Litvish folks I live amongst.
Feel free to ask further questions in the comments....