CO2

This is correct, but you could also say "Carbon Dioxide" instead.

Table 1 shows how much CO2 each of the five countryies emitted per personcapita from 1980 to 2010.

If you can't type CO2 with the small "2" below the normal text, I would use "carbon dioxide" instead.

"per person" isn't wrong, but the phrase "per capita" is usually used in data and statistics. It means the same thing, but implies the "per person" value has been calculated -- like an average.

Another way to write this could be:
"Table 1 shows CO2 emissions per capita for each of the five countries from 1980 to 2010."

Japan is the only country in this chart which discharged more CO2 in 2000 than it did in 1980 and 1990, although it emitted less in 2010 than in 2000.

I'm not sure if you mean: "more in 2000 than in 1980 and 1990 combined" (2000 > 1980+1990), or if you mean "more in 2000 than 1980 or 1990" (2000 > 1980 and 2000 > 1990).

The amount was smallest among the five countries until 2000.

I would be a little more specific about "the amount." Do you mean cumulative CO2 emissions (1980 through 2000), or the yearly emissions?

However, CO2 emissions byfrom four other countries decreased year by year, and in 2010, the UK has the smallest number.

You can replace "year by year" with "yearly" if you'd like, but it's okay as it is.

Again, I would be a little more specific with "the smallest number," especially as this is your conclusion sentence.

You could say "in 2010, the UK has the least per capita CO2 emissions."