In truth, it’s a matter of personal preference, it is essential that it contains fluoride however. For adults 1450ppm (parts per million) of fluoride is optimal and children 1000ppm is best.
Other ingredients - the addition of triclosan/copolymer has been proven to be better than basic fluoride containing toothpastes for patients with gum disease.
If you suffer from mouth ulcers you might find that they improve if you avoid toothpastes containing sodium lauryl sulphate, a foaming agent to which many people are allergic.
Sensitive toothpastes tend to contain potassium saline, stannous fluoride and arginine. They work by blocking dentinal tubules thus insulating the tooth or by polarising the tooth surface which encourages the fluoride to be more effective.
Whitening toothpastes work through abrasion, making teeth look whiter by removing stains, however this is better done by a hygienist or dentist when providing a scale and polish. Long term use of whitening toothpastes can lead to increased sensitivity and mild toothwear. If you want whiter teeth have you considered tooth whitening?