Twemoji
Emojis for everyone
Twitter’s open source emoji has you covered for all your project's emoji needs. With support for the latest Unicode emoji specification, featuring 3,245 emojis, and all for free.
Completely free. (Yes, you read that correctly.)
Multiple image formats: SVG, PNG
Supports the latest Unicode version
Simple licensing for commercial & non-commercial use

JavaScript
Uniform emoji rendering across all browsers and operating systems.

Designers
Grab all emojis as vector artwork and use them for any commercial or personal project.
Requirements
Attribution
As an open source project, attribution is critical from a legal, practical and motivational perspective in our opinion. The graphics are licensed under the CC-BY 4.0 which has a pretty good guide on best practices for attribution.
However, we consider the guide a bit onerous and as a project, will accept a mention in a project README or an 'About' section or footer on a website. In mobile applications, a common place would be in the Settings/About section (for example, see the mobile Twitter application Settings->About->Legal section). We would consider a mention in the HTML/JS source sufficient also.
© Licensing
Copyright 2020 Twitter, Inc and other contributors
Code licensed under the MIT License: http://opensource.org/licenses/MIT
Graphics licensed under CC-BY 4.0: https://creativecommons.org/licenses/by/4.0/