A vector logo is what the designer originally sends to you (Most likely in pdf).

You can check if it is a vector by zooming in, a vector logo will never be pixelated - Doesn't matter how much you zoom in.

Below are the formats that the client can look for:

.ai: Short for Adobe Illustrator, this file is commonly used in print media and digital graphics, such as logos.

.eps: Encapsulated PostScript is an older type of vector graphics file. .eps files don’t support transparency in the way more modern file formats like .ai do.

.pdf: The Portable Document Format is built for the exchange of documents across platforms and is editable in Adobe Acrobat.

.svg: The Scalable Vector Graphics format is based in XML (a mark-up language used widely across the Internet that's readable by both machines and humans). It’s useful for the web, where it can be indexed, searched and scripted.