There are several different separation standards which ATC attempts to maintain between aircraft. Note that only one of the standards below needs to be maintained; if one is valid that is sufficient. I'm answering for USA-land as you mention the FAA in your question.
Aircraft may be separated vertically. If one aircraft is directly above a point on the earth and is at an altitude of 4000 feet, and another aircraft is directly above the same point at an altitude of 6000 feet, they are separated.
IFR aircraft must be separated from other IFR aircraft by at least 1000 feet. At or above FL290 this increases to 2000 feet if either aircraft is not Reduced Vertical Separation Minima-capable (RVSM), and above FL410 the minimum is 2000 feet no matter what (and sometimes more for certain military or supersonic flights).
Within Class B, Class C and TRSA airspace, IFR aircraft are separated from VFR aircraft by at least 500 feet. This also applies to known VFR aircraft operating in the associated Class C outer area.
Within Class B and TRSA airspace, VFR aircraft are also separated from other VFR aircraft by at least 500 feet. VFR aircraft must receive an explicit clearance into Class B airspace, allowing them to be separated from other VFR aircraft. In the case of a TRSA this only applies to aircraft requesting radar services.
Lateral Radar Separation
Vertical separation is not terribly interesting. Let's instead look at lateral separation, where the two aircraft are at the same altitude, so it would not be good if they were also above the same point on the earth.
The absolute bare-bones basic standard of separation is: Did the planes hit each other? This is a highly undesirable situation. For VFR aircraft operating in Class E/G airspace, this is the only minimum that matters, and pilots will apply it themselves using the "see and avoid" principle: They see other aircraft and maneuver to avoid hitting them. This is also the only separation standard that ATC maintains between two VFR aircraft in Class C/D/E airspace, and in fact separation isn't truly lost until the planes trade paint. (Then it's a bad day for everyone.)
At a slightly higher level there is target resolution. This can be applied between VFR-VFR or VFR-IFR aircraft operating in any type of airspace, except in Class Bravo when one aircraft weighs more than 19,000 lb or is a turbojet. Target resolution means the controller issues instructions, traffic advisories, and safety alerts to ensure that the radar targets on their screen are "resolved," which is to say they do not ever touch each other. There can be some error in the positioning of the targets on the screen but if two aircraft are very near each other the difference between the two errors is likely to be minimal—so if the displayed targets never touch, you can be sure the aircraft themselves certainly don't.
Getting into defined radar separation:
- In Class Bravo airspace, VFR aircraft are separated from IFR/VFR aircraft weighing more than 19,000 lb by 1.5 nautical miles.
- In the terminal environment (i.e. TRACONs, and enroute facilities at FL230 and below where certain requirements are met) IFR aircraft are separated by 3 NM. This can be reduced to 2.5 NM on final approach, subject to various extra rules and procedures.
- If the aircraft is more than the prescribed distance from the radar antenna, or if the system indicates a need, or above FL230, IFR aircraft are separated by 5 NM.
- At or above FL600 separation increases to 10 NM between IFR aircraft.
Passing or Diverging Separation
Terminal facilities are authorized to discontinue the use of radar separation and fall back on target resolution if aircraft are on tracks which will never meet (either because one target has passed the projected track of another, or if they are opposite-direction because the two targets have passed each other). Enroute facilities can do this as well but have more onerous rules.
Wake Turbulence Separation
Smaller aircraft operating behind heavier aircraft (defined based on maximum takeoff weight) are afforded increased separation, ranging from 4 to 10 NM instead of 3 or less.
Almost invariably, using non-radar separation will provide a greater actual distance between aircraft than using radar separation, so it is not usually applied in areas of radar coverage. For myself I don't know much about it, having never applied it.
Aircraft may be separated by assigning them to fly on different airways, or different radials or arcs of navigational beacons, or other such procedures.
Aircraft operating along the same airway, or at the same airport, may be separated based on time and speed.
Airborne separation may be reduced below the applicable minimum if visual separation is used. Some other form of approved separation must exist before and after the application of visual separation, and visual separation is not allowed above FL180 (i.e. in Class A airspace) or if the leading aircraft is a super weight class.
Pilot-applied visual separation may be used if the pilot of one aircraft reports another aircraft in sight and confirms that they will maneuver to maintain their own separation (i.e. they don't hit). Tower-applied visual separation may be used if the tower controller can see both aircraft and issues instructions to ensure they don't hit.
Visual separation is not prohibited if the weather is below VFR minima (IMC) but will likely have limited usefulness because some other form of approved separation must exist once the pilot or controller loses sight of the aircraft. Some facilities have it written in their local procedures that tower-applied visual separation is assumed to be in effect whenever the weather is better than certain minima.
Not as relevant to your question but provided for completeness. Runway separation is flight-rules agnostic; the controller will apply it regardless if a flight is VFR, SVFR, or IFR. Based on the category of aircraft (category being defined in relation to engine type and number, and for single-engine propeller planes also weight) and type of operation (arrival-arrival, arrival-departure, departure-arrival, departure-departure) ATC will ensure 3000, 4500, 6000 or "the entire runway's worth" feet between aircraft operating on the same runway. They will also ensure an aircraft or vehicle does not cross a runway another aircraft is operating on.
So to finally answer your question: Pilots do not apply "separation standards" in any type of airspace, except that they are required to avoid hitting other aircraft if they see them in time. Instead, ATC applies separation minima which change based on the type of flight rules, type of aircraft, and airspace in which the aircraft is operating. In order to make sure these separation minima are maintained, pilots are required to comply with ATC clearances and instructions, or else have a really good reason why not.