A standard arrival procedure (STAR) will have multiple entry fixes (called "transitions" because that's where you transition from the en route phase to the terminal phase) that converge on a single fix for which the STAR is named, usually with one or more (optional) holds along the way that ATC can use to maintain separation. There may be more shared fixes or radar vectors after the convergence point to various IAPs. This phase is typically handled by approach control (APP), but it will be done by Center in areas without a TRACON. Note that there will usually be multiple STARs for a given airport depending on the direction you're coming from, and typically one complete set uses RNAV and another uses VORs.
Each instrument approach procedure (IAP) covers how to get to a particular runway with a certain kind of equipment; there is usually more than one IAP for a given runway, and some IAPs cover multiple runways. Regardless, once you're established on the IAP, APP (or Center) will hand you to TWR, or to CTAF if there is no TWR.
A departure procedure (DP, formerly SID) is roughly the same process in reverse: TWR hands you off to departure control (DEP), which takes you from the runway (via radar vectors or specific fixes) to a common fix and then out to diverging transition fixes, and then you're handed off to Center for en route flight. Like STARs, there is usually one set using RNAV and another using VORs.
The point of all this complexity is to safely manage a large number of aircraft, possibly flying in IMC in relatively close proximity to each other, by using paths that are known to be safe and relatively efficient yet with relatively little ATC interaction per flight--and most potential conflicts can be resolved simply by putting one of the flights in a charted holding pattern. If every flight had to be given every turn/descent/etc. by ATC, the workload would be completely unmanageable.
In an ideal world, you'd fly direct from your departure transition fix to your arrival transition fix, but in reality, you may be stuck on airways in between or at least some intermediate fixes due to ATC computer limitations.