What are the problems with wheel alignment?

Wheels aren't pointing the right way in relation to each other. Plain and simple as that.

If the wheels are aligned, there is no problem with alignment. However if they get out of alignment you can get get little to major steering "drift" and also cause improper wearing on your tires making them to wear unevenly, and usually at a faster rate than expected.

If you hit a curb for instance and knock one wheel out of alignment, you may see the steering will not  go straight when driving straight, you may see noticeable steering to one side when you let go of the steering wheel or may even feel loose or shaky when driving.

Inner and outer tie rods are adjusted so that wheels turn in or out to match the straight forward alignment when the car steering wheel is centered in the straight position. Some places will charge you 70-100 bucks for this as a one off service, others may throw it in for free with a new set of tires.

Having your alignment out of spec not only causes the drift, but like I said, causes wheel wear that depending on how out of alignment it is could even drastically cut the life of your tires... which in turn has the possibility of costing you more money then the alignment... specifically with higher cost tires.