Mainly, MIMO channels are correlated. Assuming we have 4 antennas in BS and another 4 in UE side, so we have 4*4 path between the BS and UE, these 16 paths effect on each other, i.e., they are correlated somehow. Of course, this is the description in single UE scenario, while in multi user case, it goes more complicated. To sum up, correlated channels mean paths in this channel impact on each other, meanwhile independent channel means power of certain path doesn't increase, decrease, or effect on other paths.
In two correlated channels a parameter you are interested in, say received amplitude, varies in a related way. Assuming the channels are statistically stationary with time (rarely true) and the amplitudes are bipolar with a zero average then the long-term average of the product of the amplitudes is not zero in correlated channels. This correlation can be positive or negative but if it is nonzero this channel parameter is correlated between the two channels. For independent channels, the long-term average of the product of the parameters will be zero, assuming they are zero average, and the statistics are time stationary.