Ex-center was 51-100 as Bulls coach
Cartwright's 15 seasons as an NBA player included stints in Chicago and New York and an All-Star selection in 1980.
"Bill brings a wealth of NBA experience to our coaching staff," said Nets CEO Rod Thorn.
Cartwright, 47, was Bulls head coach for parts of the past three seasons, compiling a record of 51-100.
Before that, he spent six seasons as an assistant in Chicago, including when the Bulls won the NBA championship in 1997 and 1998.
Cartwright was a first-round draft pick (third overall) of the Knicks in 1979. He played eight seasons in New York before going to Chicago in a trade for Charles Oakley in June 1988.
Terms of the deal were not disclosed.
The Celtics selected the 6-foot-8, 240-pound forward with the 40th pick in the June draft.
Reed played at Mississippi, where he averaged 18.5 points and 7.5 rebounds per game last season and was named to the All-SEC first team.
Brewer averaged 2.5 points, 1.3 assists in 12.3 minutes in a 13-game stint with the Indiana Pacers last season. He has played a total of 36 games in three years in the NBA.
Information from The Associated Press was used in this report.