References will be to FAA JO 7110.65, the "ATC Bible" which tells us how to work airplanes.
Arrivals
The baseline standard for aircraft separation is 3NM lateral (5–5–4b) or 1000' vertical (4–5–1a). When aircraft are on final approach they're descending to the runway and you can't use vertical, only 3NM in-trail.
At some airports succeeding aircraft on final may be permitted to reduce to 2.5NM provided certain conditions are met (5–5–4i).
However, in good weather ATC may use visual separation (7–2–1) in lieu of standard radar separation. If visual separation is used then the only thing remaining is same-runway separation: For turbojet aircraft, the #1 lander has to be "clear of the runway" before the #2 lander is over the runway threshold (3–10–3a1). Depending on aircraft speeds, the existence of high-speed runway exits, etc, this can translate to as little as 2NM separation when the #1 lander is at the threshold. Sometimes even less if the #2 guy slows waaaaay down.
Departures
Again, the baseline is 3NM separation. But if aircraft are assigned courses which diverge by at least 15º this can be reduced to 1NM (5–8–3a). If both are flying RNAV SIDs only 10º divergence is required (5–8–3c).
If visual separation is used then same-runway separation for two departures will be (if both are turbojet aircraft) 6000'-and-airborne (3–9–6a). Note that 6000' is one nautical mile anyway.
Departure-and-Arrival
Now things get really interesting. This is where the major delays start happening, and this is where an airport with parallel runways has a huge advantage over a single-runway airport.
The "normal" procedure is for ATC to provide visual separation between the arrival and the preceding departure. The only thing ATC has to do is ensure same-runway separation which (again for turbojets) is 6000'-and-airborne when the arrival reaches the landing threshold (3–10–3a2).
Depending on the types of aircraft involved, the speeds on final, braking action on the runway, etc, etc, the Tower will generally need somewhere around a 3.5- or a 4-mile gap between arrivals in order to safely get a departure out while maintaining 6000'-and-airborne. More will be needed if the departing aircraft is a heavy A330; less will be needed if the departure is a Lear 75. But broadly speaking 4NM is a good safe gap; nobody in the tower cab will sit up and start making concerned comments if you issue a line-up-and-wait instruction when an arrival is four miles out.
But if ATC cannot provide visual then the rule is, again, 3NM separation at all times. Actually there is yet another rule: separation may be decreased to 2NM at the time the departure starts rolling, if it will increase to 3NM within one minute after takeoff (5–8–4). Even so, this is twice the required separation in the visual scenario: 12000' instead of 6000'.
Again I want to be clear that there are a lot of different variables. But as a rule of thumb, when the visual is not available and we need to ensure "2 increasing to 3" separation we need a gap that is about 1NM more than usual. Perhaps a little less. So now instead of 4NM between arrivals we need 4.8 or 5NM.
This is why an airport like SFO can be much more efficient than SAN. At SAN you have to wait until the #1 arrival has reached the runway before putting the departure in position, and you have to wait to issue the takeoff clearance until that arrival has cleared the runway. At SFO you can put the departure in position on the parallel at any time and then clear them as soon as the #1 arrival has touched down.
That means Approach can keep sending the arrivals in at minimum spacing and Tower is still able to launch departures.
Edit: ODO
There is a delay program into SAN again tonight, and reading the advisory I see that they are running Opposite-Direction Operations. These come with another set of stringent separation requirements (JO 7210.3 2–1–38), which will generally be: the departure has to be established in a turn that will prevent conflict before the arrival reaches a ten-mile final.
That is a very onerous requirement if there's any kind of volume at all. They are mitigating it somewhat by bringing in a bunch of arrivals to Runway 9, then holding arrivals (or vectoring them) to create a large gap in order to get several departures off Runway 27. But even with that mitigation their arrival rate is severely reduced: If they're using the same runway for arrivals and departures they can work 24 inbounds per hour while still getting departures out, even down to Low IMC when using Runway 9. But when they are running ODO the rate goes down to 16 per hour.