For the New Jersey Democratic Party, the most important political geographical area is the County.
The County Committee is made of 2 members (one male and one female) for each town precinct. One of its most important responsibility is to weigh in and support the candidates to State, Federal and local races.
The members of the various municipal committees in a county make up the County Committee. The committee members are elected every 2 years during the Primaries elections (usually in June, not in the November General Election).
Each County Committee may have different rules and different level of influence in vetting, encouraging, discouraging, support and funding of candidates.
While for practical reason some key decisions are taken by the Committee chair (both at county and municipal level), every committee is eventually ran by the committee members.
The Party committees are organized by their bylaws which should be public.
In New Jersey, most county committees have the mechanism of the "party line" which in principle is like a strong endorsement but in practice it has a stronger effect that every resident should know and the registered Democratic voters have the choice to influence. NJ Salem County Democratic Committee is one with more democratic rules: No Party Line and Secret Vote to elect officials.
The best way to influence the party processes, structure, organization and outcomes is to be part of your local municipal committee (which means also be part of the county committee).
Any registered democratic voter can run to represent their voting precinct (your town, ward, district).
It is best to discuss a membership with your committee chairperson. To run for committee member the 2019 deadline is April 1st to file the petition with 20 signatures. This is an example of petition.
Here you can find a couple of slides with some graphical details on NJ most common local party structure.
If you are interested in learning more about the New Jersey "Party Line" process you can read a few articles on the Good Government Coalition NJ as well as follow some presentations.
From the same organization you can watch this short video encompassing some of the NJ Democratic practices.