How do you add UI inside cells in a google spreadsheet using app script?

Status 2018:

There seems to be no way to place buttons (drawings, images) within cells in a way that would allow them to be linked to Apps Script functions.

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This being said, there are some things that you can indeed do:

You can...

You can place images within cells using IMAGE(URL), but they cannot be linked to Apps Script functions.

You can place images within cells and link them to URLs using:
=HYPERLINK("http://example.com"; IMAGE("http://example.com/myimage.png"; 1))

You can create drawings as described in the answer of @Eduardo and they can be linked to Apps Script functions, but they will be stand-alone items that float freely "above" the spreadsheet and cannot be positioned in cells. They cannot be copied from cell to cell and they do not have a row or col position that the script function could read.

The apps UI only works for panels.

The best you can do is to draw a button yourself and put that into your spreadsheet. Than you can add a macro to it.

Go into "Insert > Drawing...", Draw a button and add it to the spreadsheet. Than click it and click "assign Macro...", then insert the name of the function you wish to execute there. The function must be defined in a script in the spreadsheet.

Alternatively you can also draw the button somewhere else and insert it as an image.

More info: https://developers.google.com/apps-script/guides/menus