In rotations, rolling without slipping is the phrase we always hope we see in a problem. Why is this good? Because we have relationships between linear and rotational motion quantities, such as v = Rw for the linear and angular speeds, or a = R(alpha) for the linear and angular accelerations. We also have the added benefit that the friction creating the torque on a rolling object is effectively a static friction, and no heat is produced.
These features no longer are true when the IS slipping. This adds unknowns to the problem, since we no longer can connect the two motions together. What do we do? Well, other information must be given to solve the problem. Perhaps the friction force is given in a rolling problem. Or the tension is given in a yo-yo problem. These are not normally given in no slipping problems.
The bottom line is we set the problems up exactly like we always do, with F = ma for linear motion and torque = I*alpha for rotational motion. We just need to solve them separately, and not as a system. This video will show two examples, so hopefully some of the fear of slippage will be taken away! Check it out.
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