is january 20 capricorn or aquarius?

It could be either, and the ONLY way to know is to find out the EXACT time you were born, and enter that, along with your full birthdate and the location of your birthplace, into the program of any website that calculates birthcharts for free.  I recommend you use either alabe.com or cafeastrology.com.

Your "Sign" is ONLY the Sign that Sun appeared to be moving through when you were born. Since this depends on where Earth is in HER orbit around Sun, and since or calendar is based on Earth's orbit .. it is very easy to find your Sun sign using just the month-day of birth. 

But the dates are only vague guidelines. (*astronomical reason why given at end, if you are curious)

Some years Sun leaves Capricorn and enters Aquarius SOMETIME during January 19th, and other years sometimes during January 20th. It is never the same from one year to another.

And your Sun sign is ONLY 2% of your entire chart.

Sun is only 1 out of 10 "planets" that your inner personality

And everyone has ALL 12 zodiac signs in their own birthchart .... it is just that if you had a planet moving through that particular sign when you were born, that sign will be a stronger influence on you.

Sun-sign astrology exists because Sun-signs are easy to find and writers make money writing about the. Sun-sign astrology is not really astrology, so don't bother with it.  Either do up the INDIVIDUAL chart, and examine ALL 40 to 50 factors in it ... or drop fake Sun-sign "astrology" altogether

* Our calendar is only 365 whole days, but Earth takes 365 and 1/4 days to complete one orbit around Sun. That is why we day every 4 years to keep our calendar matched up to Earth's orbit (Feb 29th, Leap Year).  

This also means that the date/time when Sun moves from one Sign into the next Sign WILL vary also.