The GPL only triggers when you distribute a program that is derivative of the GPL'ed program, e.g. by linking to it. You are free to run a GPL-licensed program without having to accept the license.
If the GPL'ed program is client-side JavaScript, then you are distributing the GPL'ed code and possibly derivative code to your visitors. You are only allowed to do this in accordance with the GPL. Whether your website is a derivative work of the GPL'ed code depends on what this code is and how you are using it.
Simple example: you have a single-page web-application. Your JS code uses a GPL'ed widget or library. Therefore, your SPA falls under the GPL and you are required to make your source code available in accordance with the GPL.
If you are calling a separate executable where that executable is subject to the GPL, you are very likely not creating a derived work.
Example: Your server invokes the GNU wget
command line tool to archive some web page. Since this is an entirely independent program from your code, your server code is not affected by the GPL.
If you are linking with GPL'ed code dynamically or statically then you can run this combined work, but may only distribute/publish it under the terms of the GPL. Deployment to servers under your control is not distribution.
Example: Your server-side code builds upon a GPL'ed library. Therefore, your server-side code is a derived work of the GPL'ed code. If you publish your program, you can do so only under the terms of the GPL. Running a publicly available server is not the same as publishing the server program, so you have no further restrictions.
Note that the GPL contains a family of licenses. Different versions of the GPL have different specific details and terminology. But they all share the same spirit. The LGPL licenses are a more liberal variant of the GPL, e.g. they allow you to link to LGPL'ed code under certain circumstances. The AGPL license gives end-users of web applications similar freedoms to end-users of normal applications. When running AGPL'ed software this answer does not apply as you are subject to additional requirements.