The words Madden-Julian Oscillation (MJO) refer to a tendency of the weather in some regions of the tropics to fluctuate between rainier and drier states a few times over the course of a season. In the rainy state, the wind close to the earth’s surface tends to blow from west to east (“westerly”), while in the dry state it blows from east to west (“easterly”). A given state tends to last a month or two, and to cover a region of size several thousand km in the east-west direction. Changes from rainy to dry (or vice versa) tend to progress slowly from west to east so that – for example - if it is rainy in one place, it will probably become rainy soon in places nearby to the east of there. The MJO most strongly affects the weather over the tropical Indian and western Pacific oceans, but also influences places far from there, including at higher latitudes.
In the scientific literature you will find statements like this. “The MJO is a mode of natural variability in the tropical climate system characterized by planetary spatial scale, intraseasonal (30-60 day) time scale, and eastward propagation.” Translating that into English one bit at a time:
“A mode of natural variability” has two parts, “mode of variability” and “natural”. “Natural” means “not caused by humans”. The MJO would be there in the absence of human perturbations to the climate. It is not a result of global warming (though it is quite possible that some of its properties will change as the climate warms). “Mode of variability” means that it is a tendency for some aspects of the climate to change in a particular ways. The changes are generally back and forth between two states, rather than permanent. In this case the aspects that change most dramatically include rain, clouds, and wind.
“In the tropical climate system” means that the MJO occurs primarily in the tropics; “climate system” means in this context that both the atmosphere and ocean are involved (although for the MJO we believe that the most important action is in the atmosphere, while the ocean mainly responds to that).
“Planetary spatial scale” means that if a given state of the MJO (i.e., rainy or dry, or in jargon, “active” or “suppressed”) is in place somewhere, the same state tends to be in place everywhere around there for a long distance. “Long” here means at least thousands of km. Really the term “planetary spatial scale” means “not many times smaller than the circumference of the earth”. The circumference of the earth at the equator is 40,000 km, so planetary scale means maybe at least 5,000-10,000 km.
“Intraseasonal (30-60 day) time scale” means that 30-60 days is the typical “period”, or time it takes for a given state of the MJO (rainy or dry) to repeat itself after having first switched to the other state. The time to switch from one state to the other would then be half that period. The range 30-60 days, rather than just a single number, indicates that the switching back in forth is not that regular. In mathematical terms we would say the MJO is not truly “periodic”, but only “quasi-periodic”. (This is typical of nearly all natural climate and weather fluctuations, and is what makes them difficult to predict far in advance; truly periodic fluctuations are the diurnal cycle (day-night) and the seasonal cycle, both of which are easy to predict because their periods are known exactly and never change.) The word “intraseasonal” refers to the fact that 30-60 days is a little shorter than a season (especially in tropical regions where one might say there are really only two seasons in a year) so that within a given season there is time for a few MJO cycles.
“Eastward propagation” means that if a rainy MJO phase is happening somewhere, it will happen a little later in places east of there. In particular, MJO events tend to start in the Indian ocean and move across Indonesia to the western Pacific. The speed at which they move is around 5 meters per second. After reaching the western Pacific, MJO events continue moving eastward, into the eastern Pacific, then across the Americas into the Atlantic; but in those parts of the world the change in the weather that accompanies the MJO is often not as strong as it was over the Indian and western Pacific oceans.