The possibility of a positive cosmological constant was raised by Einstein in the more than ninety years ago, and from 1998 to the present, nothing more complicated has been found to be necessary to the observed acceleration in the rate of the expansion of the Universe. The constancy of this effect everywhere in the universe has since been confirmed by a completely independent methodology related to galactic cluster growth rates in 2008.
The rate at which this happens, called the Hubble Constant, is 73.8 (plus or minus 2.4) km/sec/megaparsec.
OK, so what does that mean?
We see galaxies rushing away from us. Moreover, the farther away they are, the faster they appear to be moving. The rate of that expansion is what was measured. If you find a galaxy 1 megaparsec away (about 3.26 million light years), the expansion of space would carry it along at 73.8 km/sec (fast enough to cross the United States in about one minute!). A galaxy 2 megaparsecs away would be traveling away at 147.6 km/sec, and so on.*
* In reality it’s a little more complicated than that. Due to gravity of galaxies and clusters, the rate of expansion doesn’t really kick in until you get out to a distance of a handful of megaparsecs. But the concept is the same.
We have an abundance of riches when it comes to other theories that could also explain their observations without a cosmological constant, because the observations show a quite simple empirical relationship. Dark energy (with a matter-energy density of about roughly 10^−29 grams per cubic centimeter) or a highly even and diffuse neutrino condensate of the same density, in the universe, are equivalent to a positive cosmological constant within the equations of the relativistic law of gravity to the level of precision that our empirical observations permit. The scientific effort that has bloomed to explain their discovery is related to the mechanism that causes the effect that they observe, not to its effects or predictions.
There is more discussion the Bad Astronomy blog.
The Dark Matter Question Contrasted
While the accelerating expansion of the Universe has multiple perfectly satisfactory explanations, we have the opposite problem in the case of "dark matter" which observations of galactic rotation curves have made clear we need since 1934 since in Newtonian gravity with no dark matter, gravity would be insufficient to bind the stars at the rotating fringes of the galaxies to the galaxy and they would instead fly off into deep space.
Sixty-five years later, we have lots of theories to explain these phenomena too, but none are really satisfactory.
Cold dark matter, which was in vogue as the dominant mainstream paradigm for a couple of decades has increasingly been shown to be a poor match to the empirical evidence.
Warm dark matter posits a dark matter particle that high energy physics seems to have pretty much ruled out the existence of in precision experiments.
Some modified gravity theories fail to work in the full range of experimental data, aren't relativistic, or have been pierced by the bullet cluster example. The most promising modified gravity theories, like Moffat's MOG theory, haven't received enough attention from the mainstream physics community to determine if the tires have been kicked enough to know that it really works, and unlike General Relativity, their complexity isn't so nearly a consequence of a few general principles. Modified gravity approaches that are consistent with General Relativity in the strong field limit (there have been several permutations), generally seems to require a field with a scalar, vector and tensor component, rather than only the tensor component found in General Relativity, or the somewhat more complex scalar-tensor theories of gravity which are currently indistinguishable from General Relativity to the limits of existing experimental tests.
Inaccuracies in our measurements of the amount of baryonic matter in the universe and the our calculation of general relativistic effects of complex galactic and larger scale systems also make it unclear how much dark matter we really need (the estimates you've seen for the amounts of matter and dark matter proportionately in the universe start with a total matter number and then back out estimates of the amounts of baryonic matter from star census and interstellar gas amount estimates).
Thus, while the observations of Saul Perlmutter, Brian P. Schmidt, and Adam G. Riess had an explanation almost immediately that required only the slightest tweak of canonical fundamental physics, understanding dark matter remains a fundamental problem in physics.