vagrant is a manager for vms. it lets you manage vms (by default on virtualbox, but you can use other vm hosts too).
docker is a manager for linux containers. linux containers are linux only and a bit like chroot on steroids - they use the same linux kernel as the host, but with a different file system. so they're like lightweight, linux-only, vms. they're so lightweight that they're being "sold" as containers for single applications (so you then use docker like a package management service, installing postgres, for example, as a docker container - often this can use a separate vm as the host, so that you're not making changes to your main machine, or because you're using a mac).
fig is new to me but seems to be a wrapper around docker that simplifies some things.
the above is a simplification - for example, i'm pretty sure vagrant these days can also manage docker images (linux containers + docker metadata).
vagrant is a manager for vms. it lets you manage vms (by default on virtualbox, but you can use other vm hosts too).
docker is a manager for linux containers. linux containers are linux only and a bit like chroot on steroids - they use the same linux kernel as the host, but with a different file system. so they're like lightweight, linux-only, vms. they're so lightweight that they're being "sold" as containers for single applications (so you then use docker like a package management service, installing postgres, for example, as a docker container - often this can use a separate vm as the host, so that you're not making changes to your main machine, or because you're using a mac).
fig is new to me but seems to be a wrapper around docker that simplifies some things.
the above is a simplification - for example, i'm pretty sure vagrant these days can also manage docker images (linux containers + docker metadata).