Just south of downtown.
The North border is The 10 Freeway
The West border is The 110 Freeway (if you hit USC, you went too far)
The East border is technically Central, but most folks don't trip if you extend it to Alameda*
The South border is really Vernon, but Vernon touches South Park and a lot of folk include South Park which extends to Slauson. Anything south of Slauson is where Watts starts.
*those who do trip off the East boundary do so because East of Alameda is where it really starts to get Mexicaney (for lack of a better word). For example, when Ice Cube dissed Cypress Hill on "King Of The Hill" he said:
"I hear you claiming South Central, wait/
you ain't from my 'hood, y'all hoes from South Gate!"
The City of South Gate is quite close to the South Central area but is not a part of Los Angeles Proper, and borders on Alameda.