How much “wiggle room” (forgive me for not knowing the scientific term) in terms of landing site are we allowed? For example: we want the Mars probe to land at this spot on the Martian surface. We consider it a success if we land within, say, 100km? 5km? 1km?
If we wanted to send someone up to the Moon to clean up all the equipment left there from the Apollo missions, would we be able to get a craft to within a few hundred meters?
Curiosity landed about 1.5 miles from its target location, which was considered very good indeed. Apollo 12 landed only 600 feet from its target, so I don’t think there would be any problem getting a craft to within a few hundred meters today.
If we put in the effort, we can get extremely close. We probably could literally land on a dime, if we wanted to. Depending on the mission, we might be more or less flexible in where we land, though, and if we don’t need to land on a dime, then there’s no point going to the effort and expense to ensure that.
600 feet is less than a few hundred meters. 
Not in metric meters. 
Point there, but I was actually referring to the landing on Mars, rather than on the Moon. 1.5 miles is a bit more than a few hundred meters.
The term you are looking for is “tolerance” or “variability”.
The answers so far are essentially correct. It is worth noting that it is easy to make an landing on an airless body with low gravity as the only considerations are purely ballistic, and so the only real sources of variability are changes in the local gravity field and errors in the navigation, guidance and autopilot systems. With a body that has some atmosphere the vehicle is also subject to aeroloads and and drag which both degrades the inertial solution due to perturbations and causes additional errors due to differences between the aero model built into the guidance code. Of course, an atmosphere also offers the oppotunity for aerodynamic controlled flight to correct errors at lower speed.
Mars is actually the worst solid body in the solar system to land upon becuase it has just enough atmosphere to be troublesome but not enough to effectively glide without a gigantic lifting surface. Getting the MSL to a precise landing was the result of an enormous multidisciplinary effort and iterations of design, coupled fluid-structural interaction simulations, and wind tunnel testing.
Stranger
Is a Mars landing made by a vehicle that makes it’s own course corrections for the landing? It would seem that the time lag between instrumentation and commands from Earth would cause problems. But then it is all Newtonian. 
Indeed. Signal travel time is around 5 minutes when Mars is close to Earth, and can be more than 20 minutes. Double those times for a control signal from Earth to respond to something reported by a Mars lander.
I don’t know anything about this and I bow to your supreme wisdom on the subject, Stranger, but I would have thought Venus would have been the most difficult due to it’s violent atmosphere. If that’s not the case, ignorance is definitely fought on this!
Wouldn’t Venus be even harder to land on than Mars? Yeah, there’s that nice thick atmosphere to work with, but an atmosphere isn’t as helpful when it’s melting your wings.
And even on the Moon, one other variable might be that you don’t know your landing spot as well as you thought. Armstrong and Aldrin had to burn right up against the limits of their fuel reserves, because the planned landing site had too many boulders for a safe landing.
<nevermind>
In the vein of this subject, this bimonthly issue of the AIAA Journal of Spacecraft and Rockets (Vol 31, No 3, May-June 2014) has a special section on supersonic retropropulsion specifically focused on crewed Mars landing missions with seven papers covering numerical simulation, wind tunnel testing, and data analysis. The lead paper, “Development of Supersonic Retropropulsion for Future Mars Systems” is a survey of the current state of the art in design, simulation, and testing of retropropulsion systems (e.g. using rockets to slow and control the descent of multi-ton vehicles in the tenuous but aerodynamically significant atmosphere of Mars) with an estimated schedule and necessary advances to bring the technology readiness level (TRL) from the current 2-3 up to the point that it could be tested in upper Earth atmosphere to simulate Martian aerocapture and descent for 40+ ton payload (TRL 6). The abstract is provided below:
Recent studies have concluded that Viking-era entry system deceleration technologies are extremely difficult to scale for progressively larger payloads (tens of metric tons) required for human Mars exploration. Supersonic retropropulsion is one of a few developing technologies that may enable future human-scale Mars entry systems. However, in order to be considered as a viable technology for future missions, supersonic retropropulsion will require significant maturation beyond its current state. This paper proposes major milestones for advancing the component technologies of supersonic retropropulsion such that it can be reliably used on Mars technology demonstration missions to land larger payloads than are currently possible using Viking-based systems. The development roadmap includes technology gates that are achieved through ground-based testing and high-fidelity analysis, culminating with subscale flight testing in Earth’s atmosphere that demonstrates stable and controlled flight. The component technologies requiring advancement include large engines (100s of kilonewtons of thrust) capable of throttling and gimbaling, entry vehicle aerodynamics and aerothermodynamics modeling, entry vehicle stability and control methods, reference vehicle systems engineering and analyses, and high-fidelity models for entry trajectory simulations. Finally, a notional schedule is proposed for advancing the technology from suborbital free-flight tests at Earth through larger and more complex system-level technology demonstrations and precursor missions at Mars.
The article goes on to discuss the various EDL architectures sturdied in the NASA Mars Design Reference Architecture 5.0 (DRA5, which I have not worked with and am only passingly familiar with) and note that the EDL Systems Analysis team identified Option #1 (aerocapture and shrouded hypersonic reentry followed by supersonic retropropulsion with powered landing) as being the preferred architecture from the standpoint of simplicity and minimal payload mass to delivery a 40t vehicle to the Martian surface. (Of the eight architectures in the study, four use supersonic retropropulsion, and the only one of the remaining four that is significantly lighter than option #1 requires staged inflatable annular decelerators from aerocapture through transonic regimes which will likely degrade landing accuracy significantly.)
The fundamental conclusions of the paper are that the current aerocapture/aerobraking technology used for Viking, Pathfinder, MER, and MSL (blunt conical aeroshells) is at about the limits that it can be scaled up, the technology to land a vehicle capable of sustaining even a small crewed mission to the surface of Mars is in a nascent state, and will take at least a couple of decades before both the simulation and ground test capability is sufficient to develop a system which is ready to be tested in actual flight conditions.
Another one of the papers, “Analysis of Navier-Stokes Codes Applied to Supersonic Retropropulsion Wind-Tunnel Test” has some very interesting studies on the effects of the plume on the supersonic flowfield and shock boundary including aft turbulence at high thrust coefficients which could potentially destabilize the vehicle. The paper also admits the limitations of current tools and modeling techniques which don’t fully simulate unstable three dimensional flow and don’t including the chemical kinetics which will add further thermodynamic considerations into the design of any retropropulsive vehicle designed for a Mars landing.
Stranger