There are several 'types' of memory in Firefox.

Type about:preferences#privacy<enter> in the address bar.

The button next to History, select Use Custom Settings.

Turn off Always Use Private Browsing Mode Turn on Remember My Browsing And Download History At the bottom of the page, turn on Clear History When Firefox Closes. At the far right, press the Settings button. Turn on ONLY Cache and Form And Search History leaving the others off.

They could be search engine suggestions that are retrieved in real time as you type your query. These do not have a special symbol; items from Firefox search history have a little "clock" icon next to them.

You can turn off search engine suggestions on the Options page. This article has the details: Search suggestions in Firefox.

Was that it?

jscher2000 said

They could be search engine suggestions that are retrieved in real time as you type your query. These do not have a special symbol; items from Firefox search history have a little "clock" icon next to them. You can turn off search engine suggestions on the Options page. This article has the details: Search suggestions in Firefox. Was that it?

Sort of. Why do some suggestions have a greyed out hat-looking thing next to them?

Vanderbilt21 said

Why do some suggestions have a greyed out hat-looking thing next to them?

Is this in the drop-down below the address bar? Those are matches among your open tabs. If you click one of those, Firefox will switch to showing the open tab instead of re-opening that address in the current tab. In fact, usually instead of showing the address it will say Switch to tab. (You can hold down the Shift key to override Switch to tab.)