How can I connect my phone without a HDMI port to a non-MHL TV? I have an older LG p990 that has a HDMI port, and with the right cable I am able to show movies on it on the TV.

My new phone is a Nexus 4, it has no HDMI port. Based on this question and answer, I know how to connect it to an MHL compatible TV. But how can I use a TV without MHL capability? Is there an adapter for that? How is it called, where can I buy one?
MHL compatibility on a TV only means that it will supply sufficient power (500mA, 900mA, or 2000mA, depending on which version of the MHL spec it is compatible with) through the HDMI port to power the adapter and charge the phone.

An MHL adapter will work just fine with a non-MHL TV.  You just need to supply your own power from a regular charger.  This is what the female microUSB port on the MHL-to-HDMI adapters is for.  You plug the charger into that, plug the HDMI cable into the TV, and MHL cable into the phone and you're off to the races.

Not all phones support MHL however, the Nexus 4 in the question supports SlimPort instead.