To be somewhat more specific than @geoff that gave an overview, regarding MOXIE (Mars OXygen In situ resource utilization Experiment), according to brief notice in the NASA press release that accompanied the Mars 2020 Rover scientific payload announcement:
The Mars Oxygen ISRU Experiment (MOXIE), an exploration technology
investigation that will produce oxygen from Martian atmospheric carbon
dioxide.
And a mock-up of the presumed layout of these announced scientific experiments and equipment:
An artist concept image of where seven carefully-selected instruments will be located on NASA’s Mars 2020 rover. The instruments will conduct unprecedented science and exploration technology investigations on the Red Planet as never before. Image credit: NASA
More detailed information will slowly trickle into the press as specifics are decided upon during the course of next few years, and indeed SpaceRef just published a story on it (I'd assume following media teleconference that was scheduled a few hours after yesterday's public announcement of the Mars 2020 Rover science package on NASA TV):
MOXIE -- short for Mars OXygen In situ resource utilization Experiment
-- was selected from 58 instrument proposals submitted by research teams around the world. The experiment, currently scheduled to launch
in the summer of 2020, is a specialized reverse fuel cell whose
primary function is to consume electricity in order to produce oxygen
on Mars, where the atmosphere is 96 percent carbon dioxide. If proven
to work on the Mars 2020 mission, a MOXIE-like system could later be
used to produce oxygen on a larger scale, both for life-sustaining
activities for human travelers and to provide liquid oxygen needed to
burn the rocket fuel for a return trip to Earth.
More on MOXIE in the article.
So this experiment is going to extract oxygen from the atmospheric carbon dioxide that constitutes 96% of Martian atmosphere. And the modus operandi for this particular experiment seems will be what Wikipedia describes as electrolysis of the atmosphere, possibly using zirconia as a catalyst:
$$\require{mhchem}\ce{2CO2} + \text{energy} → \ce{2CO + O2}$$
See e.g. Zirconia Electrolysis Cells for Oxygen Generation from Carbon Dioxide for Mars In-Situ Resource Utilization Applications. Additional note that carbon monoxide is merely an immediate reaction product and, while it could be used as fuel on Mars, it can also be further reduced by use of catalysts and additional energy into elemental carbon and oxygen, or by adding hydrogen by secondary Bosch reaction to graphite and water, and so on. Since it's not yet known exactly how MOXIE will do that, and what use its reaction products will be put to, I'd reserve further judgment on it for the time being.
There are of course also other possibilities to extract oxygen on Mars, for example from possible water ice pockets embedded deep into its regolith and using electrolysis to split water into oxygen and hydrogen, from dry ice that we know exists in abundance in Martian regolith and as a precipitate above it in polar regions and then using same or similar oxygen extraction methods as MOXIE will employ, to superheating and melting oxidized minerals, rust and metallic ore to release locked oxygen in it, and so on. But for the first steps, extracting it directly from atmospheric CO2 seems to be the easiest way to do it without needing to employ a whole truckload of machinery that we can't afford to land there, or don't really know how to yet.