What does Catherine want to show by pressing the electric bell?
The general answer to the question is that Catherine wants to show that she is a high status, cultivated person. If the question refers to her specific use of the bell in Act III, the answer is that she wants to summon a servant, and show herself to be cool, powerful, and unperturbed by the controversy of the coat.
Here are the particulars.
First, Catherine wants to be counted among the most cultured, civilized people. So while her husband, Petkoff, is away at war, she has had an electric servant bell installed in her home. It's a complete novelty to Petkoff; he hasn't even heard of these devices, and asks her what it's for. Her answer clarifies what it means to her: It's an important act of conforming to ways of cultural elites.
CATHERINE. You touch a button; something tinkles in the kitchen; and then Nicola comes up.
PETKOFF. Why not shout for him?
CATHERINE. Civilized people never shout for their servants. I've learnt that while you were away.
Second, regarding Catherine's use of the bell in Act III, Catherine presses the button to summon Nicola so she can tell him to retrieve the coat from the blue closet. Using the bell allows Catherine to create a more dramatic effect -- highlighting her apparent calm and confidence. She might have called out for Nicola instead, or left the library to get the coat herself. But it seems clear that she is interested in making an impression as the cool, regal mistress of the house. After she presses the bell, and her husband questions her about it, the stage directions describe her confident, grand air:
She looks at him majestically and silently resumes her chair and her needlework.