I can't say exactly what is happening without seeing the complete Dockerfile that was used to build the image, but I am pretty certain that the trouble you are having is just because whatever task that is being started inside the container is finishing and exiting.
Docker containers work by having some command assigned (using ENTRYPOINT or CMD directives in the Dockerfile, or as an argument to docker start or docker run on the command line) which is the program that is started when the container loads. The container will live for as long as that task continues to run, and once that program finishes the container will terminate.
To specify the startup entrypoint at the command line, try:
docker create -it [image] /bin/bash
Then start it like this:
docker start -ia [Container ID]
The container will exit once the shell exits, because this is assigning the shell as the entry point.
cURL may not be installed by default. It is possible to install it using apt-get. But again, once the shell is closed, the container will stop and any changes will be lost. As a start, try creating a new directory somewhere, and then add a file called Dockerfile with this content:
FROM ubuntu:latest
RUN apt-get update && apt-get install -y curl
ENTRYPOINT ["/bin/bash"]
That will create a new image with curl installed. then, from inside the new directory where the Dockerfile was created, use:
docker build .
docker images
which will build a new image, using the Dockerfile as the blueprint.
Once the build finishes, find the image ID for the new container, and run it using:
docker run -it [image id]
Ultimately, to make Docker really useful, the typical approach is to replace that last line in the Dockerfile (ENTRYPOINT ["command"]) with something that will continue running forever (like ENTRYPOINT ["apache2"] or ENTRYPOINT ["redis"] or similar). If you have experience using regular desktop/server OS installs, and full virtual machines like VMWare or VirtualBox, just remember that Docker is very different; the way it works and the patterns used to deploy it are not the same.