Short answer: you can't. Ports below 1024 can be opened only by root.
The long answer: you can redirect connections on port 80 to some other port you can open as normal user.
Run as root:
# iptables -t nat -A PREROUTING -p tcp --dport 80 -j REDIRECT --to-port 8080
As loopback devices (like localhost) do not use the prerouting rules, if you need to use localhost, etc., add this rule as well (thanks @Francesco):
# iptables -t nat -I OUTPUT -p tcp -d 127.0.0.1 --dport 80 -j REDIRECT --to-ports 8080
NOTE: The above solution is not well suited for multi-user systems, as any user can open port 8080 (or any other high port you decide to use), thus intercepting the traffic. (Credits to CesarB).
EDIT: as per comment question - to delete the above rule:
# iptables -t nat --line-numbers -n -L
This will output something like:
Chain PREROUTING (policy ACCEPT)
num target prot opt source destination
1 REDIRECT tcp -- 0.0.0.0/0 0.0.0.0/0 tcp dpt:8080 redir ports 8088
2 REDIRECT tcp -- 0.0.0.0/0 0.0.0.0/0 tcp dpt:80 redir ports 8080
The rule you are interested in is nr. 2, so to delete it:
# iptables -t nat -D PREROUTING 2