Special VFR will allow an aircraft to depart or arrive at an airdrome that is technically IFR (below 1000 and 3), down to 1 mile visibility, in the day time.
As a professional pilot who has been doing this for 40 years, I recommend that you don't use this very often. Flying around in 1 mile of vis is really not wise. Say the reported vis is 2.5 miles in fog, you are instrument rated, and you just want to get out of town. This would be OK. Why instrument rated? Because the 2.5 miles in fog, can, with a small temperature or wind shift, become something way less than a mile, and without an instrument rating you are going to have to declare an emergency, and try to get down with your fanny intact. I know it was forecast to become CAVU, but you are no less dead.
To get it you simply call the controlling agency, usually the control tower. Example, "XXX Tower, Cessna 3347G 10 miles South, with information Whiskey, requesting Special VFR for landing."
The specific regulation that covers this is:
§ 91.157 Special VFR weather minimums.
(a) Except as provided in appendix D, section 3, of this part, special VFR operations may be conducted under the weather minimums and requirements of this section, instead of those contained in §91.155, below 10,000 feet MSL within the airspace contained by the upward extension of the lateral boundaries of the controlled airspace designated to the surface for an airport.
(b) Special VFR operations may only be conducted—
(1) With an ATC clearance;
(2) Clear of clouds;
(3) Except for helicopters, when flight visibility is at least 1 statute mile; and
(4) Except for helicopters, between sunrise and sunset (or in Alaska, when the sun is 6 degrees or more below the horizon) unless—
(i) The person being granted the ATC clearance meets the applicable requirements for instrument flight under part 61 of this chapter; and
(ii) The aircraft is equipped as required in §91.205(d).
(c) No person may take off or land an aircraft (other than a helicopter) under special VFR—
(1) Unless ground visibility is at least 1 statute mile; or
(2) If ground visibility is not reported, unless flight visibility is at least 1 statute mile. For the purposes of this paragraph, the term flight visibility includes the visibility from the cockpit of an aircraft in takeoff position if:
(i) The flight is conducted under this part 91; and
(ii) The airport at which the aircraft is located is a satellite airport that does not have weather reporting capabilities.
(d) The determination of visibility by a pilot in accordance with paragraph (c)(2) of this section is not an official weather report or an official ground visibility report.