What is an icy rock that has bright tail when it comes close to the sun called?

You're fishing for "comet", but the setup of your riddle is defective.

-- A comet need not necessarily be in orbit around the sun.

Its appearance may be a once-in-eternity event.

-- A comet only has a 'tail' during the small portion of its orbit

when it's relatively near the sun.

-- The whole notion of a "tail" is misleading. That particular whatchamacallit

always points away from the sun, so when the comet has rounded the sun

and is headed out again, the so-called 'tail' goes in front of it.