Bullet Designs

Bullet designs have to solve two primary problems. They must first form a seal with the gun's bore. The worse the seal, the more gas, generated by the rapid combustion of the propellant charge, leaks past the bullet, reducing the efficiency. The bullet must also engage the rifling without damaging the gun's bore. Bullets must have a surface which will form this seal without causing excessive friction. What happens to a bullet inside the bore is termed internal ballistics. A bullet must also be consistent with the next bullet so that shots may be fired precisely.
Once it leaves the barrel, it is governed by external ballistics. Here, the bullet's shape is important for aerodynamics, as is the rotation imparted by the rifling. Rotational forces stabilize the bullet gyroscopically as well as aerodynamically. Any asymmetry in the bullet is largely cancelled as it spins. With smooth-bore firearms, a spherical shape was optimum because no matter how it was oriented, it presented a uniform front. These unstable bullets tumbled erratically, but the aerodynamic shape changed little giving moderate accuracy. Generally, bullet shapes are a compromise between aerodynamics, interior ballistics necessities, and terminal ballistics requirements. Another method of stabilization is for the center of mass of the bullet to be as far forward as practical as in the minnie ball or the shuttlecock. This allows the bullet to fly front-forward by means of aerodynamics.
See Terminal ballistics and/or Stopping power for an overview of how bullet design affects what happens when a bullet hits something, and how this is affected by its design. What happens to the bullet is dictated as much by what it hits and how it hits, as by the bullet itself (just like how its interaction with air was critical in external ballistics). Bullets are generally designed to penetrate, deform, and/or break apart. For a given material and bullet, which of these happens is determined especially by the strike velocity.
Actual bullet shapes are many and varied, and an array of them can be found in any reloading manual that sells bullet moulds. RCBS, one of many makers, offers many different designs, starting with the basic round ball. With a mould, bullets can be made at home for reloading one's own ammunition, where local laws allow. Hand-casting, however, is only time- and cost-effective for solid lead bullets. Cast and jacketed bullets are also commercially available from numerous manufacturers for hand loading and are much more convenient than casting bullets from bulk lead.