Element is a web UI library with versions for Vue, React and Angular. Element is quite massive including a lot of different components, it has almost all the components you need. Document is also quite easy to read, another point I found interesting is that it is quite suitable for building admin pages to manage.
Buefy is an open source library built on top of Vue.js and Bulma. All components are responsive across device screens and do not depend on other libraries. Bulma is a mobile first CSS framework using Flexbox. I quite like Buefy because it is beautiful and lightweight, although not as complete as Element is, but still very useful.
Bootstrap provides the most comprehensive experience with Boostrap 4 components and Grid system for Vue.js 2.4+, complete with automatic and extensible WAI-ARIA access, so if your project is using it bootstrap then you can think of using bootstrap-vue, which includes everything Bootstrap 4 has and is built to use with Vue.js.
Vuetify is a semantic component framework for vuejs. Vuetify is designed according to the documentation and has a clean user interface. The number of supporting components is also quite good. However, you should consider it in projects where the UI is not document-oriented because customizing it is also tiring.
Quasar is a front-end framework based on Vue. You should check it out if you plan to work on responsive web and mobile apps. It is a framework that supports features like minification and caching - in addition to components for your framework.
Quasar is probably the biggest of all the Vue UI component libraries I mentioned here. It has detailed documentation supporting end-to-end execution.