Actually, depending on the instructor, there can be several levels to partner-practice, including sparring (after all, developing reaction time without simulating real situations would be exceedingly difficult).
Generally, as I understand it, the process goes: Solo Form; Tui Shou (Push Hands); Da Lu (Large Rollback); San Shou (Dispersing Hands - partner form); Free Fighting. How and to what degree each is emphasized usually depends on the traditions of the instructor's school.
A good introduction to the process of learning the martial art of Taijiquan is in "The Power of Internal Martial Arts" by B.K. Frantzis.