Most of the following discussions are about reaction drives - those that work by throwing something out the back to push a space craft forwards. All require 'fuel' (actually, reaction mass, but it's the same thing for chemical rockets and fuel is easier to write), of which any vehicle will have a finite supply. This means that any reaction drive will have a maximum speed - though it can keep on accelerating, eventually it runs out of fuel. For spacecraft, this is generally known as Delta V (dV) - the maximum change in velocity possible.
A ship with a Delta V of 20km/s could accelerate up to 5km/s in one direction, coast for a bit, then turn around and decelerate by 5km/s to come to a stop, for a total of 10km/s. It then has enough fuel to accelerate back up to 5km/s back towards home and stop again using up the last of its fuel. Though it could get up to 20km/s by burning all its fuel, doing so would leave it with no more fuel to stop or manoeuvre. A missile might do this, but a space craft with crew on board probably wouldn't.
Acceleration is another factor that must be considered, and is the one most people think of when considering space ship drives. Acceleration is how quickly you can get from not moving to moving fast. A ship with an acceleration of 1g (9.8m/s/s, though we'll assume 10m/s/s for simplicity) can go from a standing start to 36km/s in an hour. This does assume that the ship has enough dV to accelerate this long.
Note that the Space Shuttle could accelerate at several g's, but that was only for a few minutes. It's total dV was in the order of 10km/s - 90% of which was needed just to get into Low Earth Orbit (LEO) to dock with the space station. Deceleration was managed by atmospheric drag, since it didn't have enough fuel to do much more than reduce it's orbit back into the atmosphere.
Chemical Rockets (TL 6+)
Chemical rockets are the simplest form of space propulsion, and are assumed to be possible from TL 6 onwards. On Earth, they didn't start being used until TL 7, but the technology was there if there had been the necessary will. Technically, the basic physics was understood in TL 3 or 4, but the ability to build a big enough rocket to actually get into space would come along much later.
The problem with chemical rockets is that they are limited to chemical energy, which greatly limits their dV. Even at TL 8 or 9, dV for rocket ships will be limited to a few 10s of km/s.
Nuclear Fission Drive (TL 7+)
These are similar to chemical rockets except that they use a nuclear reactor to heat the reaction mass. As such, they have much greater efficiencies, but can be very dirty. Though they can be built using TL 7 technology, they are often ignored at this point due to the difficulties of radiation shielding and of building a nuclear engine light enough to be practical.
Fission drives have a better delta-v than chemical rockets, but lower thrust.
Nuclear Pulse Drive (TL 7+)
Similar to other nuclear drives at this point, pulse drives suffer from the problem of being potentially highly dangerous due to radioactive contamination. Pulse drives work by throwing small nuclear bombs out the back and detonating them - the energy from each explosion pushes the spacecraft forward.
Though they sound utterly insane, prototypes suggest that the work, and they can provide massive dV and acceleration, allowing travel across the solar system in a matter of weeks. Most cultures don't develop pulse drives at this stage unless there is a very real need for them.
Ion Drives (TL 8+)
Ion drives are a form of electric propulsion that uses some energy source to propel a reaction mass out the back at high speed. These tend to be very mass efficient, having a high Delta V, but also suffer from having very low accelerations. They are useful over long distances, but not so useful around a planet.
Plasma Drive (TL 9+)
These are the first drives to allow a practical form of space travel. They are lighter and more efficient than the nuclear engines of TL 7, but work in a similar way. A nuclear reactor is used to heat a reaction mass (normally hydrogen or water) to a plasma, and ejects it out of the rear. They are relatively clean, since the reactor elements don't come into contact with the propellant.
Using water as a propellant is cheaper and simpler than hydrogen, but not as efficient.
High Performance Plasma Drive (TL 10+)
These are a refinement on the standard plasma drive, and provide a significantly higher thrust and a modest increase in efficiency. This greatly reduces the size and weight of the drive itself. They also provide a variable thrust mode, which enables them to run at higher efficiency but lower thrust, which is well suited for interplanetary travel.
Plasma Torch (TL 12+)
The plasma torch is the end point of nuclear plasma drives, providing a much higher efficiency over previous designs. Similar to the high performance drive, the plasma torch has variable thrust enabling it to take advantage of high thrust in combat or whilst in orbit, but using a much more efficient cruising mode for interplanetary travel.
Anti-Matter Drive (TL 13+)
Ignoring the slightly problem of shielding, and their tendency to explode if damaged, anti-matter drives are the best reaction drives available.