In 1899, physicists Ernest Rutherford (working in McGill University in Montreal, Canada) and Paul Villard (working in Paris) separated radiation into three types: eventually named alpha, beta, and gamma by Rutherford, based on penetration of objects and deflection by a magnetic field. Alpha rays were defined by Rutherford as those having the lowest penetration of ordinary objects.
Rutherfords work also included measurements of the ratio of an alpha particles mass to its charge, which led him to the hypothesis that alpha particles were doubly charged helium ions (later shown to be bare helium nuclei). In 1907, Ernest Rutherford and Thomas Royds finally proved that alpha particles were indeed helium ions. To do this they collected and purified the gas emitted by radium, a known alpha particle emitter, in a glass tube. An electric spark discharge inside the tube produced light. Subsequent study of the spectra of this light showed that the gas was helium and thus the alpha particles were indeed the helium ions.
Because alpha particles occur naturally, but can have energy high enough to participate in a nuclear reaction, study of them led to much early knowledge of nuclear physics. Rutherford used alpha particles emitted by radium bromide to infer that J. J. Thomsons Plum pudding model of the atom was fundamentally flawed. In Rutherfords gold foil experiment conducted by his students Hans Geiger and Ernest Marsden, a narrow beam of alpha particles was established, passing through very thin (a few hundred atoms thick) gold foil. The alpha particles were detected by a zinc sulfide screen, which emits a flash of light upon an alpha particle collision. Rutherford hypothesized that, assuming the "plum pudding" model of the atom was correct, the positively charged alpha particles would be only slightly deflected, if at all, by the dispersed positive charge predicted.
It was found that some of the alpha particles were deflected at much larger angles than expected (at a suggestion by Rutherford to check it) and some even bounced almost directly back. Although most of the alpha particles went straight through as expected, Rutherford commented that the few particles that were deflected was akin to shooting a fifteen-inch shell at tissue paper only to have it bounce off, again assuming the "plum pudding" theory was correct. It was determined that the atom's positive charge was concentrated in a small area in its center, making the positive charge dense enough to deflect any positively charged alpha particles that came close to what was later termed the nucleus.
Prior to this discovery, it was not known that alpha particles were themselves atomic nuclei, nor was the existence of protons or neutrons known. After this discovery, J.J. Thomsons "plum pudding" model was abandoned, and Rutherfords experiment led to the Bohr model and later the modern wave-mechanical model of the atom.
In 1917, Rutherford went on to use alpha particles to accidentally produce what he later understood as a directed nuclear transmutation of one element to another. Transmutation of elements from one to another had been understood since 1901 as a result of natural radioactive decay, but when Rutherford projected alpha particles from alpha decay into air, he discovered this produced a new type of radiation which proved to be hydrogen nuclei (Rutherford named these protons). Further experimentation showed the protons to be coming from the nitrogen component of air, and the reaction was deduced to be a transmutation of nitrogen into oxygen in the reaction
:N + α → O + p 
This was the first discovered nuclear reaction.
To the adjacent pictures: According to the energy-loss curve by Bragg, it is recognizable that the alpha particle indeed loses more energy on the end of the trace.