I would like to know the difference between marginal VFR (MVFR) and Special VFR (SVFR), particularly when flying in the US airspace. What does each of them mean exactly?
Straight out of the Aviation Weather Services advisory circular (AC 00-45G with changes 1 and 2):
Marginal Visual Flight Rules (MVFR) indicated on the Weather Depiction Chart represents ceiling 1,000 to 3,000 feet and/or visibility 3 to 5 statute miles and VFR operations can take place. MVFR areas are outlined with a solid line, but the area is not shaded. MVFR areas are typically shaded blue in colorized versions of the chart.
For all regulatory purposes, this has no rule changes from VFR. I believe the reason it exists is for pilots to more easily identify areas where the weather conditions are approaching IMC.
Special VFR in the US
Special VFR allows a pilot to fly within the lateral boundaries of controlled airspace to the ground with at least 1SM visibility and clear of clouds.
Special VFR is something that can be granted by ATC upon request by a pilot. It may only be granted within the lateral boundaries of controlled airspace that goes to the ground AND the airport must be reporting IFR conditions at the surface.
If the special VFR request is being made at night, the pilot and airplane must be IFR legal.
So if an airport within class D airspace to the surface is reporting 3SM and a 1,500ft overcast ceiling, that is MVFR (VFR). Therefore, special VFR cannot be approved by ATC because the field isn't reporting IFR.
If that airport's conditions were an 800ft overcast ceiling and 2SM, then they could approve it.
MVFR is an advisory term.
Special VFR is an, on request by pilot only, "required weather minimums" change that may be approved by ATC.
For the purposes of special VFR in the US, here is the FAA regulation. (14 CFR 91.157)