Political parties impact policies through electing candidates that have specific preferences for legislation.
The majority political party fills leadership roles in the House & Senate. For example, the current Speaker of the House is Paul Ryan (R.) following Trump's ascension to President. Nancy Pelosi had been the Speaker under Obama's presidency. This allows the party to push for legislation to be approved through the House and Senate. The President also hopes to be able to appoint judges for life. President Trump has appointed TWO justices-- Neil Gorsuch and Brett Kavanaugh.
Parties also have an impact through the redistricting process though they have to be careful with this process as it can lead to federal courts determining that their redistricting was unconstitutional, see Shaw v Reno.
Party Structures:
National Chairs:
Hill Committees:
State and Local Parties: