There are the IAF and the FAF, which has clear roles. One is where you start the approach and the other where you start to descent and end the approach. But what is the purpose of an Intermediate fix?
The Pilot/Controller Glossary can shed some light on your question.
The GPS approach is the clearest example of an Intermediate Approach Fix.
IF/IAWP− Intermediate Fix/Initial Approach Way- point. The waypoint where the final approach course of a T approach meets the crossbar of the T. When designated (in conjunction with a TAA) this waypoint will be used as an IAWP when approaching the airport from certain directions, and as an IFWP when beginning the approach from another IAWP.
INTERMEDIATE APPROACH SEGMENT [ICAO]− That segment of an instrument approach procedure between either the intermediate approach fix and the final approach fix or point, or between the end of a reversal, race track or dead reckoning track procedure and the final approach fix or point, as appropriate.
The Instrument Flying Handbook warns that, “Randomly joining an approach at an intermediate fix does not ensure terrain clearance."
On other approaches, there may not be an intermediate fix, but the segment is defined as follows:
INTERMEDIATE FIX− The fix that identifies the beginning of the intermediate approach segment of an instrument approach procedure. The fix is not normally identified on the instrument approach chart as an intermediate fix (IF).
INITIAL APPROACH SEGMENT [ICAO]− That segment of an instrument approach procedure between the initial approach fix and the intermediate approach fix or, where applicable, the final approach fix or point.
The Instrument Flying Handbook also talks about intermediate fixes where there is sometimes a charted Intermediate Fix:
When a teardrop procedure turn is depicted and a course reversal is required, unless otherwise authorized by ATC, this type of procedure must be executed. [Figure 1-15] The teardrop procedure consists of departure from an IAF on the published outbound course followed by a turn toward and intercepting the inbound course at or prior to the intermediate fix or point. Its purpose is to permit an aircraft to reverse direction and lose considerable altitude within reasonably limited airspace. Where no fix is available to mark the beginning of the intermediate segment, it shall be assumed to commence at a point 10 NM prior to the FAF. When the facility is located on the airport, an aircraft is considered to be on final approach upon completion of the penetration turn. However, the final approach segment begins on the final approach course 10 NM from the facility.
Holding in Lieu of Procedure Turn
A holding pattern in lieu of a procedure turn may be specified for course reversal in some procedures. In such cases, the holding pattern is established over an intermediate fix (IF) or a FAF.
In addition to all the elements in JScarry's excellent answer, intermediate fixes can also be used to step down altitudes, such as during the ILS 10C at Chicago O'Hare.