The word 'halo' comes from the Greek halos, which means the ring of light shown around the sun.
The Sun Cross was probably an early representation of that oldest and most powerful god - the sun1. No surprise therefore that the symbolism transferred into other religions such as Hellenistic Greek and Roman religions, Hinduism and Buddhism. And arguably the most prolific sacred art with haloes has been Christian art. Not only has the bright circle been adopted, but also the original term 'halo' has been retained.
The photo on the right shows a halo around the depiction of Jesus Christ. When a cross is shown as part of the halo, as in this picture, the image is nearly always that of Jesus Christ. Without a cross, one can assume the image is of a saint, an angel, a prophet or some other holy person2.
A circle or arc is much easier to draw or carve than an image of a holy person, therefore just the halo is sometimes shown on its own to represent a holy person. (See for example, the halo on the lamb in Angus Dei artwork.)
A Halo Cross is a generic term for many crosses with such shapes. The Celtic Cross (above, left) is perhaps the most common and a few more examples are shown here on the right.
The halo shape can be secondary to some other sacred entity. The Rainbow, for example reminds us of God's promise to mankind and also represents light, the rainbow after the storm of living in this world. The Ankh reminds us of the rebirth possible through Jesus' sacrifice; the Crown of Thorns worn by Jesus when he was crucified; the Wreath, made from leaves, plucked from bushes and trees, sacrificed to make the wreath, conveniently in a halo shape; and the Enshrined cross, acknowledging the importance of the cross itself.
"The righteous shall shine forth as the Halo in the kingdom of their Father." (Matt. 13:43)
Halos are not restricted to Christianity; other religions such as Buddhism, Hinduism and Jainism use halos in iconography.
The Japanese Buddhist statue on the left is shown with a kouhai or gokou (halo), holding a trident weapon to protect babies & young children.
The Hindu god Shiva is shown here on the right with Parvati, both with a halo. Shiva has a Third Eye, a necklace made of skulls, a trident and a few snakes.