ports

netstat prints network connections, routing tables, interface statistics, masquerade connections, and multicast memberships.

Using the parameter -l (or --listening) it will show only listening sockets/ports (which are omitted by default.).
--numeric-ports shows numerical port numbers but does not affect the resolution of host or user names (e.g. instead of showing the name ssh, it will show the value 22).

We used netstat using the following syntax to check which sockets/ports are open on the current machine:

netstat --listening --numeric-ports;

The results appeared as follows:

[george@bytefreaks ~]$ netstat --listening --numeric-ports
 Active Internet connections (only servers)
 Proto Recv-Q Send-Q Local Address           Foreign Address         State      
 tcp        0      0 0.0.0.0:22              0.0.0.0:*               LISTEN     
 tcp        0      0 localhost:25            0.0.0.0:*               LISTEN     
 tcp6       0      0 [::]:44300              [::]:*                  LISTEN     
 tcp6       0      0 [::]:8080               [::]:*                  LISTEN     
 tcp6       0      0 [::]:22                 [::]:*                  LISTEN     
 tcp6       0      0 localhost:25            [::]:*                  LISTEN     
 udp        0      0 0.0.0.0:39925           0.0.0.0:*                          
 udp        0      0 0.0.0.0:24186           0.0.0.0:*                          
 udp        0      0 0.0.0.0:68              0.0.0.0:*                          
 udp        0      0 localhost:323           0.0.0.0:*                          
 udp        0      0 0.0.0.0:5353            0.0.0.0:*                          
 udp6       0      0 localhost:323           [::]:*                             
 udp6       0      0 [::]:33848              [::]:*                             
 udp6       0      0 [::]:61453              [::]:*                             
 raw6       0      0 [::]:58                 [::]:*                  7          
 Active UNIX domain sockets (only servers)
 Proto RefCnt Flags       Type       State         I-Node   Path
 unix  2      [ ACC ]     STREAM     LISTENING     22489    public/showq
 unix  2      [ ACC ]     STREAM     LISTENING     22445    public/pickup
 unix  2      [ ACC ]     STREAM     LISTENING     22449    public/cleanup
 unix  2      [ ACC ]     STREAM     LISTENING     22477    private/proxymap
 unix  2      [ ACC ]     STREAM     LISTENING     22480    private/proxywrite
 unix  2      [ ACC ]     STREAM     LISTENING     15452    /run/systemd/private
 unix  2      [ ACC ]     STREAM     LISTENING     22483    private/smtp
 unix  2      [ ACC ]     STREAM     LISTENING     22486    private/relay
 unix  2      [ ACC ]     STREAM     LISTENING     22492    private/error
 unix  2      [ ACC ]     STREAM     LISTENING     22495    private/retry
 unix  2      [ ACC ]     STREAM     LISTENING     22498    private/discard
 unix  2      [ ACC ]     STREAM     LISTENING     22501    private/local
 unix  2      [ ACC ]     STREAM     LISTENING     22504    private/virtual
 unix  2      [ ACC ]     STREAM     LISTENING     22507    private/lmtp
 unix  2      [ ACC ]     STREAM     LISTENING     22510    private/anvil
 unix  2      [ ACC ]     STREAM     LISTENING     22513    private/scache
 unix  2      [ ACC ]     STREAM     LISTENING     14445    /var/run/NetworkManager/private-dhcp
 unix  2      [ ACC ]     SEQPACKET  LISTENING     15476    /run/udev/control
 unix  2      [ ACC ]     STREAM     LISTENING     1404     /run/systemd/journal/stdout
 unix  2      [ ACC ]     STREAM     LISTENING     22452    public/qmgr
 unix  2      [ ACC ]     STREAM     LISTENING     15498    /run/lvm/lvmpolld.socket
 unix  2      [ ACC ]     STREAM     LISTENING     22474    public/flush
 unix  2      [ ACC ]     STREAM     LISTENING     22471    private/verify
 unix  2      [ ACC ]     STREAM     LISTENING     16034    /var/run/dbus/system_bus_socket
 unix  2      [ ACC ]     STREAM     LISTENING     16037    /var/run/avahi-daemon/socket
 unix  2      [ ACC ]     STREAM     LISTENING     15537    /run/lvm/lvmetad.socket
 unix  2      [ ACC ]     STREAM     LISTENING     22456    private/tlsmgr
 unix  2      [ ACC ]     STREAM     LISTENING     22459    private/rewrite
 unix  2      [ ACC ]     STREAM     LISTENING     22462    private/bounce
 unix  2      [ ACC ]     STREAM     LISTENING     22465    private/defer
 unix  2      [ ACC ]     STREAM     LISTENING     22468    private/trace

Check a specific port if it is open from a remote machine

In case you want to check a specific port if it is open from a remote machine, you can use nmap.
Using nmap to scan specific ports allows you to check if a remote machine appears to have open ports available to you.
nmap is a network exploration tool and security / port scanner.

The following example checks ports 80 and 8080 on 192.168.1.199 if they are open.

[george@bytefreaks ~]$ nmap -vv -p 80,8080 192.168.1.199
 
 Starting Nmap 6.40 ( http://nmap.org ) at 2017-02-22 14:10 EET
 Initiating Ping Scan at 14:10
 Scanning 192.168.1.199 [2 ports]
 Completed Ping Scan at 14:10, 0.00s elapsed (1 total hosts)
 Initiating Parallel DNS resolution of 1 host. at 14:10
 Completed Parallel DNS resolution of 1 host. at 14:10, 0.00s elapsed
 Initiating Connect Scan at 14:10
 Scanning 192.168.1.199 [2 ports]
 Discovered open port 8080/tcp on 192.168.1.199
 Completed Connect Scan at 14:10, 0.00s elapsed (2 total ports)
 Nmap scan report for 192.168.1.199
 Host is up (0.000060s latency).
 Scanned at 2017-02-22 14:10:29 EET for 0s
 PORT     STATE  SERVICE
 80/tcp   closed http
 8080/tcp open   http-proxy
 
 Read data files from: /usr/bin/../share/nmap
 Nmap done: 1 IP address (1 host up) scanned in 0.03 seconds

The -vv parameter for nmap increases the verbosity of the results.
The -p parameter defines the ports to be checked.