A bifurcated trial is a trial which will take place in two separate phases, dependent on the outcome of the first portion.
In a civil matter, the first portion determines whether a defendant is liable issue in a matter which can involve contract or tort issues. If the defendant is found liable, the trial proceeds to the second phase where damages are determined.
In a criminal matter, the first portion determines the defendant's guilt or innocence. If found guilty, the second phase determines punishment. These are most common in felony cases, especially when capital punishment is a potential sentence. Misdemeanor cases usually proceed in one phase, where both guilt/innocence and punishment (if found guilty) are determined at the same time.