The answer to this varies for each of the branches. The people are most directly involved with the legislative branch and least involved with the judicial branch.
The people are directly involved with the legislative and executive branches in that they vote for the people who are the leaders of those branches. In addition, the people can pressure these branches through the activities of interest groups or simply through writing letters expressing their opinions.
None of these options is really open with the judicial branch. The people are only able to be involved with this branch through bringing law suits. Sometimes interest groups will have legal strategies that involve bringing lawsuits to challenge laws that they wish to overturn.
In these ways, the people can be involved to some extent in each of the three branches, but they are least involved with the judicial branch.
Public participation is very important especially in running a democratic government and this does not stop after elections. The public is responsible for most of the actions and decisions that the government makes through expressing their opinions and suggestions.
The public can influence the legislature by directly communicating their ideas to members of congress. During debates the public participates as individuals or groups to either support or oppose passage of a new bill through correspondence to their representatives. For example, the voting in of strong Net Neutrality rules by the FCC after direct participation by millions of Americans.
In a similar way, the public can participate in the running of the executive through petitions aimed at specific departments or communications directly to the White House. For instance, one can visit the We the People link on the White House website to create a petition, participate in open petitions and view responses from the government on the petitions. On the same website members of the public can directly reach the President through letters to the White House.
Public participation in the Judiciary is different as explained by Pohnphei. Members of the public can bring law suits challenging some policies or actions by the government. The public can also follow proceedings in the courts. The jury composed of members of the public is another example of how the public participates in the judiciary.