Yes, you can. You can save your progress like in the real NES. Just define a save folder, where a .sav file will be created by the FCEultra and compatible with many others emulators.
That file will have the same name as your rom, for example, if you're playing "Super Mario Bros 3 (U).nes" or .zip, , the save/battery file will be named "Super Mario Bros 3 (U).sav"