Situation when there is strong linear relationship among predictor variables, so that their correlation matrix becomes (almost) singular. This "ill condition" makes it hard to determine the unique role each of the predictors is playing: estimation problems arise and standard errors are increased. Bivariately very high correlated predictors are one example of multicollinearity.
Multicollinearity refers to when predictor variables are (linearly) correlated with each other. Although the term is sometimes used to mean perfectly correlated (i.e., $r=1$) only, it is more often used to simply mean strongly correlated. Multicollinearity need not be manifested in bivariate correlations; a variable could be correlated with several other variables such that all bivariate correlations are low.
Conceptually, the existence of multicollinearity means that it is difficult to determine the role each of the correlated variables is playing. Mathematically, it manifests in larger standard errors. Thus, collinearity reduces statistical power.
Multicollinearity can produce counter-intuitive phenomena. For example, when a collinear variable is added or dropped from a model, other variables can switch between significance and non-significance, and / or the sign of their relationship with the response can switch between positive and negative.
Additionally, when there is multicollinearity, small changes in the data can lead to large changes in the parameter estimates, even reversals of sign.
Detecting and addressing multicollinearity is an important topic in multivariable statistical modeling. Two common methods of detecting multicollinearity are variance inflation factors (VIFs) and condition indexes, the latter are preferred by Belsely (see references) in his seminal book on multicollinearity.
References Belsley, D. A. (1991). Conditioning Diagnostics: Collinearity and Weak Data in Regression. Wiley.
