The Lion is one of Buddhism's most potent symbols. Traditionally, the lion is associated with regality, strength, and power. It is therefore an appropriate symbol for the Buddha who tradition has it was a royal prince. The Buddha's teachings are sometimes referred to as the 'Lion's Roar', again indicative of their strength and power.
The image on the left shows a capital from a pillar of Asoka: the Lions of Sarnath. Sarnath is where the Buddha first preached, and these lions echo his teachings to the four quarters of the world, sometimes called 'the Lion's Roar'. The wheel symbolizes Buddhist law and also Asoka's legitimacy as an enlightened ruler.
Especially in Tibetan Buddhist art, lions are often depicted on the throne the Buddha sits on, but these are Snowlions (mythical creatures), and they actually represent the eight main Bodhisattvas (students of the Buddha).
What seems a much later development is the depiction of the Buddha's eyes (especially on stupas), as is frequently seen in Nepal. They look in all four directions, representing the omniscient mind of a Buddha.