Will not stay on when not plugged into the car outlet?

First make sure that the device is indicating that the battery is being charged or that the battery is fully charged. this is indicated on the menu screen by the battery icon as being 4 bars of full green color.
if the unit will not charge in your vehicle, you might need to replace the fuse that is located at the tip of the vehicle adapter.
1. to change the fuse unscrew the black, round end piece on the vehicle power adapter and remove it. inspect the fuse to see if it is blown. If not end here.
2. if determined the fuse is blown remove the fuse (glass and silver cylinder), and replace it with a 1A fast blow fuse.
3.ensure that the silver tip is placed in the black end piece. screw on the black end piece
if the fuse isn't blown and the unit still wont charge you may have received a usb cable with your Garmin Nuvi. If you didnt recieve one any mini usb cable should do. On this model you can connect the Nuvi to a computer using a usb cable. And see if you can charge your unit through this method. *Note* that the computer must be on for the unit to be able to be charged.
If you can charge your Nuvi unit on the computer then you may have something wrong with your cars adapter. I doubt this is the problem however if the unit is able to be turned on.
If you cannot charge the unit via a car adapter (with working fuse) or a USB cable. The most likely problem is the battery is no longer accepting a charge. On the listed unit a Garmin Nuvi 265 the battery is not user replaceable. So you would have to contact Garmin to get the battery changed or see if there is a local battery store that is able to change a battery on this model unit.
If you need further assistance please feel free to follow up with a comment and i will be here to assist you. If this assisted you please rate the comment. thanks and have a great day.