Party in the Electorate
This is the largest party of a political party, and it just means the voters. So, anyone who votes for a candidate from a particular party is considered part of the party in the electorate. In the United States, it's so easy for someone to be part of a political party. Just claim you're a member, and you're a member. In European countries, you may have to pay a fee or have a membership card, but in America, it's very simple.
Party as an Organization
The party as an organization refers to the fact that each political party has a national office, a full-time staff, and budgets. In addition to one large national office, each party has state and local headquarters. The head of that part is called the national chairperson. As of 2015, the chairman of the RNC is Reince Priebus, and the chairwoman of the DNC is Debbie Schultz.
Party in Government
The party in government refers to elected officials who represent a party. These people are the main spokespeople for their parties, and include Senators, Governors, and the President.