Which is the oldest star?
Short answer: The oldest star we know of is called "HD 140283", AKA the "Methuselah Star". It's 14.46 ± 0.8 billion years old.
Hubble Space Telescope image of the Methuselah Star
Credit: Digitized Sky Survey (DSS), STScI/AURA, Palomar/Caltech, and UKSTU/AAO
Long answer: Determining the age of stars is difficult, so there's always a degree of uncertainty when talking about which star is the oldest. Also, we can really only study individual stars that are relatively nearby in our own galaxy (the Milky Way), so there's no way to know the age of all the stars in distant galaxies.
At this stage, the star known as "HD 140283" or the "Methuselah Star" is the best candidate for the oldest known star. Located about 200 light-years away in the constellation Libra, astronomers estimate its age at 14.46 billion years plus or minus 0.8 billion years.
You may notice that 14.46 billion years is older than the estimated age of the universe (13.8 billion years), which is obviously a problem. However, the uncertainty error of 0.8 billion years does bring the Methuselah Star back into the universe's age, which itself is still an estimate with some uncertainty. Still, it's an uncomfortable situation and astronomers would dearly love to pin down both ages with more accuracy.