Molniya orbits are always associated with Russian satellites. What advantage do they have over other orbits?
Most countries use geostationary satellites for communications. They have the advantage that the satellite appears to be stationary in the sky, so you can use a simple fixed dish to communicate with it.
Russia/the former USSR has a lot of land at high latitudes. A geostationary satellite would need a lot of transmitter power to be usable there. A Molniya orbit fixes that by having the satellite at a high inclination so it has a good view of a most of Russian territory at the same time. The orbit is also highly elliptical so it spends most of its time in a good position. The drawbacks are that you need a tracking dish for communications, and you need 3 satellites to have continuous coverage.
This wiki article on the molniya orbit provides this helpful starting explanation from which one can begin to tie together both previous answers by Hobbes and Mike:
Much of the area of the former Soviet Union, and Russia in particular, is located at high latitudes. To broadcast to these latitudes from a geostationary orbit (above the Earth's equator) would require considerable power due to the low elevation angles. A satellite in a Molniya orbit is better suited to communications in these regions because it looks directly down on them.
An additional advantage is that considerably less launch energy is needed to place a spacecraft into a Molniya orbit than into a geostationary orbit. Disadvantages are that, as opposed to a spacecraft in a geostationary orbit, the ground station needs a steerable antenna to track the spacecraft and that the spacecraft will pass the Van Allen radiation belt four times per day.
So there are two immediate advantageous features that suited the former Soviet Union, namely improved elevation angles for communications and the launch energy.
To explain the latter a little more: the propellant mass required to achieve a given orbit depends in part on the difference in degrees between the latitude of the launch site and the inclination of the target orbit. For example, the lowest latitude launch site used by the former Sovient Union was at Baikonur with a latitude of 46 degrees, hence it would take least energy to go into a 46 degree orbit, more to get to 63.4 degrees and much more to get to GEO at 0 degrees.
The next point is "why choose 63.4 degrees specifically?". The same wiki article goes on to say:
In general, the oblateness of the Earth perturbs the argument of perigee, so that even if the apogee started near the north pole, it would gradually move unless constantly corrected with station-keeping thruster burns. To avoid this expenditure of fuel, the Molniya orbit uses an inclination of 63.4°, for which these perturbations are zero.
Put simply, the orbit is most stable there. To determine which feature amongst those mentioned was the driving feature when the former Soviet Union planned the Molniya communications satellite programme would be a big historical pursuit in its own right though it appears that dealing with the perturbation through the satellite life was a significant challenge as they chose 63.4 degrees. I haven't done the sums to compare the launch mass under the three scenarios for propellant and transmitter power by the way.