SELECT name FROM tbl ORDER BY weight DESC LIMIT 1
Much more performant than the other answer and results in one row only.
Use this:
select name
from tbl
where weight = (select max(weight) from tbl)
    ORDER BY DESC puts rows with null values at the top.
To avoid returning results corresponding to null values:
SELECT name FROM tbl WHERE weight = (SELECT MAX(weight) FROM tbl);
Note: This query will return multiple results if multiple people have a weight equal to the maximum weight. To grab just one, add LIMIT 1 to the end of the query.
Acknowledgements and more information:
Why do NULL values come first when ordering DESC in a PostgreSQL query?
MIN/MAX vs ORDER BY and LIMIT
Postgres MAX Function
If you need to find multiple rows, e.g. date on which each person had maximum weight:
name | weight | day
don    110      1
don    120      20
don    110      30
joe    90       1
joe    80       15
joe    85       30
i.e. for "don" you want to get "don | 120 | 20" and for joe you want "joe | 90 | 1", then you can write:
SELECT name, max(weight), (array_agg(day ORDER BY weight DESC))[1] FROM tbl GROUP BY name
    
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