|MadSci Network: Physics|
In this discussion, I'll ignore the effects of special relativity: the effects aren't important enough at 10% of the speed of light to affect my calculations. Also, this is a huge subject and my discussion of the various propulsion systems is very brief: please follow the links for more detailed explanation.
The most important concept in answering this question is the rocket equation. This equation relates the final velocity a spacecraft can reach to the amount of fuel it carries, and to the speed of exhaust gases out the back of the rocket.
The exhaust velocity, also called specific impulse, is the most important factor in the efficiency of a rocket engine. The highest exhaust velocity achievable using chemical fuel and oxidizer is 4.5 km/sec, from hydrogen/oxygen. If you plug this into the rocket equation along with a final velocity of 30,000 km/sec (10% c) and say 10 tonnes for the empty weight of the spacecraft, the fuel required is greater than the mass of all galaxies in the observed universe! We obviously can't use chemical rockets to reach 10% c.
The highest exhaust velocity yet achieved by an engine tested in space is the ion propulsion system used on Deep Space 1. This system has an exhaust velocity of 30 km/sec. Plugging this into the rocket equation gives a vastly smaller amount of fuel, but it's still more than everything in the observed universe.
There isn't really anything I would consider "under development" which can match the exhaust velocity of an ion engine. So let's look at propulsion systems which are still only suggested. (Some of these "suggestions" have a fair amount of research behind them, but there are still large gaps in our knowledge of physics preventing them from being built.)
A rocket which gets its energy from controlled nuclear fusion might be able to have an exhaust velocity of 1000 km/second. This technique would require 1014 tonnes of fuel to accelerate a 10-tonne spacecraft to 30,000 km/sec. This is still impossible: the fuel required is comparable to the mass of all the hydrogen in the water of the Carribean Sea, and you could never build a fuel tank weighing less than 10 tonnes which could hold this much material.
Finally, there's antimatter propulsion, which might produce exhaust velocities of 10,000 km/sec -- this is 3% the speed of light. Such a spacecraft would "only" need 100 tonnes of fuel. This is quite reasonable, but it remains to be seen if one can create and compactly store such a huge amount of antimatter. Also, the consequences of an accident involving this much antimatter would be (almost literally) earth-shattering.
All these are "conventional rockets" in the sense that they carry their fuel with them. This is a problem because it adds to the mass which must be accelerated, so you need extra fuel to accelerate the fuel, and even more fuel to push that fuel, and so on. There are other possibilities. For example, the Bussard Ramjet works kind of like a jet aircraft engine: it collects the thin hydrogen gas between the stars in a giant electromagnetic scoop, burs the gas in a nuclear fusion reaction, and lets the hot exhaust out the back. It's a fusion rocket which doesn't need to carry its own fuel. As such, the rocket equation doesn't apply. However, we don't yet understand how to make a self-sustaining controlled fusion reaction, and the interstellar gases aren't ideal for fueling a fusion reaction. A Bussard ramjet is limited by the density of the interstellar gas: if it's too thin, the rocket can't get enough fuel. I read somewhere (but can't find the source) that recent estimates of the gas density say that it's thin enough that a ramjet might not work.
Finally, there's the idea of a "light sail". Photons carry momentum, so when light bounces off a mirror it exerts a very slight push against the mirror, just as wind exerts a push on a ship's sail. For travel within the solar system, one could use sunlight to push the spacecraft, but for travel through the darkness between the stars, a gigantic system of lasers and highly-focused lenses is required. Such a system could reach 10% the speed of light over the course of several years, but the lasers would have to supply an amount of power comparable to the total power output of today's human civilization. Unlike fusion power, the difficulties here are engineering challenges rather than fundamental lack of understanding of the physics involved, but the engineering challenges are gigantic.
For more information on this topic, I recommend Chapter 11 of "Islands In the Sky", edited by Stanley Schmidt and Robert Zubrin, Paul Woodmansee's pages on interstellar travel, and the "Lunar Institute of Technology"'s starship design pages.
Try the links in the MadSci Library for more information on Physics.