Web: Vidd.me is like Imgur for videos. You can quickly and anonymously upload any video you want, without a need to sign up for an account. And yeah, it also supports GIFs!
As well as the webapp, it's also available on Android and iOS. You can upload a video or GIF from your hard drive or paste a link. Animated GIFs are converted into videos of the same quality but are 1/16th the file size for faster loading, the developers say.
Currently, there are no restrictions on what you can upload, so be warned that you might come across some NSFW content. Unfortunately, there's no way to embed the videos yet -- hopefully that will be added soon.