The cp command can be used to copy files.

One specific file can be copied TO the container like:

docker cp foo.txt mycontainer:/foo.txt

One specific file can be copied FROM the container like:

docker cp mycontainer:/foo.txt foo.txt

For emphasis, mycontainer is a container ID, not an image ID.

Multiple files contained by the folder src can be copied into the target folder using:

docker cp src/. mycontainer:/target
docker cp mycontainer:/src/. target

Reference: Docker CLI docs for cp

In Docker versions prior to 1.8 it was only possible to copy files from a container to the host. Not from the host to a container.