is my imac supposed to back itself up by time machine or only on external drive

As John explained, in order for Time Machine to work, it requires an external HD. Please refer to Back up your Mac with Time Machine where Apple explains this. Most experienced users on these forums recommend the follow External Hard Disk due to their high quality, ease of use, price and the service the manufacturer provides. It is recommended that you buy an External Hard Disk that has 2-3 times the capacity of the amount of data you have.

OWC Mercury Elite Pro

They are a US based company and shipping is generally free in the USA, if you are outside the USA you will find the following link helpful so you don't have to pay too much for shipping. OWC European Reseller

Time Machine requires a hard disk drive or partition dedicated for backups. Strictly speaking it does not have to be external but that's the usual method.