When ancient people looked up in the night sky, they noticed constellations (meaningful patterns of stars in the sky) and assigned these names.  These patterns were the same night to night and year to year.  They also noticed that there were points of light that moved through some of the constellations over time, changing their relationship to the patterns every night.  They called these "planets", which means, "wanderers", and they assigned these special meaning in their mythologies. They seemed to look a little different too, because the light from them didn't flicker.

People from many different cultures assigned special meaning to the constellations through which the planets wandered -- these became the constellations of the zodiac.  Your "sun sign" simply refers to where the sun was against the background of stars when you were born.  Like the planets, the sun only shows up in front of certain constellations (Leo, Gemini, etc.).  Of course, you can't see the constellations during the day, but they are there, and people figured out which ones the sun would be in front of at different times of the year.

We now understand that the reason the planets only appear to move through certain constellations is that they travel in a plane around the sun and that they are so much closer to us than the background stars. (Their relative proximity is the reason their light doesn't flicker: the beam of light from them (reflected sunlight in their case) is somewhat thicker than the beam of light from a distant star, and it is less likely to be disturbed by the atmosphere as it travels to our eye.

In the image below, which is NOT TO SCALE, Mars would be said to be "in Gemini", as our line of sight would place Mars against this constellation in the sky.  Notice how Mars would never appear in front of the Big Dipper (Ursa Major), nor would we ever calculate that the sun was in front of the Big Dipper from our perspective.  When you ask someone what sign they are, they never say, "I'm a Big Dipper" (unless they have a heavy chewing tobacco habit and didn't quite hear what you said).