After another half-life, there is 1/2 of that 1/2 left = 1/2 × 1/2 = 1/4 of original amount of the parent left.
When plants absorb carbon-dioxide in the photosynthesis process, some of the carbon dioxide has the carbon-14 atom in the molecule.
Assuming that our atmosphere's composition and the cosmic ray flux has not changed significantly in the last few thousand years, you can find the age of the organic material by comparing its carbon-14/carbon-12 ratios to those of now-living plants.
There are several ways to figure out relative ages, that is, if one thing is older than another.
For example, looking at a series of layers in the side of a cliff, the younger layers will be on top of the older layers.
Since radioactive rocks have been observed for only a few decades, how do you know you can trust these long half-lives and the long ages derived?