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Generally speaking, below the theater naval commander you’re going to have a number of different units of organization, with nothing as straightforward as Army brigades/battalions/companies/&c. The largest operational commands at the tactical level are Carrier Strike Groups (CSG), which each typically command a carrier, a cruiser, a Destroyer Squadron (DESRON) and an Air Wing (CVW, for Carrier Air Wing). The DESRON has a number of subordinate destroyers (and sometimes some frigates, though those are being rapidly retired), and the Air Wing has a number of subordinate squadrons, typically about four F/A-18 fighter squadrons, an E-2C/D airborne early warning squadron, an E/A-18G airborne electronic attack squadron, and two helo squadrons (one flying the MH-60R, the other the MH-60S).
The other large operational command is the Expeditionary Strike Group, which is similar but built around an amphibious assault ship as its flagship (really a small aircraft carrier with a well deck), and will also include a dock landing ship and amphibious transport dock, as well as a Marine Expeditionary Unit (MEU) along with its air complement.
Other types of units don’t fall under one of these large strike groups, such as submarines which are organized administratively into squadrons and then into groups, or maritime patrol aircraft which are in squadrons and wings, but will pretty much always be operating under direct operational control of the theater commander when deployed.
There is also the administrative chain of command, which is organized under Type Commanders, or TYCOMs. These are responsible for managing communities of various types of units, such as SURFOR (Surface Forces), SUBFOR (Sub Forces) AIRFOR (Air Forces) and so forth. The newest TYCOM is IDFOR (Information Dominance Forces), which is administratively responsible for the fairly new Information Dominance Corps, which comprises the intelligence, cryptology, meteorology/oceanography and information professional (i.e., IT) communities.
Confusing, I know! I’m still learning more all the time.
EDIT: Also worth mentioning that operationally, naval units are operationally attached to whatever theater they’re in at the moment. So, a Carrier Strike Group deploying from Norfolk, VA to the Arabian Gulf will cut over from US Fleet Forces (what used to be Second Fleet) to Sixth Fleet / NAVEUR at some point during their trip across the Atlantic, then cut over to Fifth Fleet / NAVCENT at some point immediately around their southbound Suez Canal transit. They’ll then repeat the same process on their way home.