Insert image or link links
There are few standard "Text formats" for a text field. Which one you actually can use depends on your permission settings (for example permission Use the Full HTML text format). 

Then you have the settings for each of the text format (admin/config/content/formats), there you can set up what options a specific format has (you can add/remove image/link buttons).

The last thing to know is that order in which text formats are listed in the administrator page is important. It decides which text format is used as a default. A typical problem with ordering of text formats are automatic messages with broken links (result of "Plain text" format having too high priority).