Well a hydrogen atom has one proton and one electron. So a proton is a hydrogen atom without an electron. Now a proton can't really exist on its own, so we usually show them bonded to some base (lone pair of electrons). Most organic chemist hydrogen and proton interchangeably.
The earliest experiments to understand the nature of acids and bases were concerned with substances that would either increase hydrogen ion (proton) H+ concentration or increase hydroxide ion concentrations in aqueous (water) solutions. These substances were termed acids and bases respectively. For example, the addition of HCl to water increases the H+ concentration in aqueous solutions. Likewise, NaOH is considered a base since it increases the -OH concentration when added to water.
At the time of these studies, the nature of H+ ions was unclear. They assumed that protons could exist as "bare naked" protons. Today we know that protons exist as hydronium ions. A bare proton is simply an empty 1s orbital. In a hydronium ion, the empty 1s orbital of the proton interacts with lone pairs of a water molecule forming a covalent bond.
The Arrhenius definition was important since it provided the concept of protons (H+) as acids and hydroxide ions (-OH) as bases. The Bronsted-Lowry definition of acids/bases provides a more complete definition in which substance are classified as acids if they donate a proton and bases if they accept a proton. This definition also introduced the concept of conjugate acids and conjugate bases. The conjugate acid or base is simply the acid or base in the reverse direction. The acid becomes a conjugate base while the base becomes the conjugate acid.
While Bronstead acids/bases definition is concerned with protons, the Lewis acid view is from the perspective of the lone pairs involved. A Lewis acid is an electron pair acceptor, while a Lewis base is an electron-pair donor. Some molecules and ions behave like protons (acids) and they don't even possess a proton. For example, boron and aluminum halides (BCl3 and AlCl3) are notorious Lewis acids. Its the electronic structure of these molecules that make them good Lewis acids. Each has an empty p orbital that can accept electron density. Remember acids are just empty orbitals or electron sinks! A proton is an empty s orbital, while BF3 has an empty p orbital (electron sink).
Just like Bronstead acid-base reactions, a Lewis acid will react with a Lewis base. In the following example, the molecule on the left donates its lone pairs (Lewis base) to an empty p orbital of BH3. This forms a Lewis Acid/Base complex as shown.