In this context, an 'open computer' is any computer that allows outside access to it without any request for authorisation, such as a username or password.
Open Relay
An open relay allows anyone to send email through a particular computer without authorisation, allowing the sender to potentially hide their details and use the open relay for malicious activity.
Open Proxy
Open proxies are similar in that they allow internet traffic (e.g. web browsing) to go through the open computer. Some computers are intended to be this way, but others are open without the user's knowledge due to misconfiguration or virus infection.
Security Implications
Computers that are operating as open relays and proxies without the user's knowledge are often used for malicious purposes such as sending unsolicited mail, spreading a virus or launching an attack on another network.