I want to make a move to Docker, so I've just started to mess around with it. I've installed Docker on a VirtualBox Ubuntu 15.10 (Wily Werewolf) installation and as suggested here I then tried running a basic nginx Docker image:

$ docker run --name mynginx1 -P -d nginx
Cannot connect to the Docker daemon. Is the docker daemon running on this host?

So I checked out whether Docker was running:

$ sudo service docker status
● docker.service - Docker Application Container Engine
   Loaded: loaded (/lib/systemd/system/docker.service; enabled; vendor preset: enabled)
   Active: active (running) since vr 2015-11-06 08:41:48 CET; 15min ago
     Docs: https://docs.docker.com
 Main PID: 7542 (docker)
   CGroup: /system.slice/docker.service
           └─7542 /usr/bin/docker daemon -H fd://
nov 06 08:41:47 kramer65-VirtualBox systemd[1]: Starting Docker Application Container Engine…
nov 06 08:41:47 kramer65-VirtualBox docker[7542]: time=“2015-11-06T08:41:47.900410966+01:00” level=info msg=“API …ock”
nov 06 08:41:48 kramer65-VirtualBox docker[7542]: time=“2015-11-06T08:41:48.033514149+01:00” level=info msg=“Fire…lse”
nov 06 08:41:48 kramer65-VirtualBox docker[7542]: time=“2015-11-06T08:41:48.141594321+01:00” level=info msg=“Defa…ess”
nov 06 08:41:48 kramer65-VirtualBox docker[7542]: time=“2015-11-06T08:41:48.416294436+01:00” level=warning msg=“Y…it.”
nov 06 08:41:48 kramer65-VirtualBox docker[7542]: time=“2015-11-06T08:41:48.565507576+01:00” level=info msg=“Load…rt.”
nov 06 08:41:48 kramer65-VirtualBox docker[7542]: time=“2015-11-06T08:41:48.567907022+01:00” level=info msg=“Load…ne.”
nov 06 08:41:48 kramer65-VirtualBox docker[7542]: time=“2015-11-06T08:41:48.567945214+01:00” level=info msg=“Daem…ion”
nov 06 08:41:48 kramer65-VirtualBox docker[7542]: time=“2015-11-06T08:41:48.567969891+01:00” level=info msg=“Dock….9.0
nov 06 08:41:48 kramer65-VirtualBox systemd[1]: Started Docker Application Container Engine.
Hint: Some lines were ellipsized, use -l to show in full.

This suggests that the Docker daemon is actually already running, but to be sure I just started the Docker daemon manually:

$ sudo docker daemon
INFO[0000] API listen on /var/run/docker.sock           
INFO[0000] [graphdriver] using prior storage driver "aufs" 
INFO[0000] Firewalld running: false                     
INFO[0000] Default bridge (docker0) is assigned with an IP address XXX.XX.X.X/XX. Daemon option --bip can be used to set a preferred IP address 
WARN[0000] Your kernel does not support swap memory limit. 
INFO[0000] Loading containers: start.                   
INFO[0000] Loading containers: done.                 

INFO[0000] Daemon has completed initialization       

INFO[0000] Docker daemon                                 commit=76d6bc9 execdriver=native-0.2 graphdriver=aufs version=1.9.0

I then tried running the image again, but with the same result:

$ docker run --name mynginx1 -P -d nginx
Cannot connect to the Docker daemon. Is the docker daemon running on this host?

I tried sudo'ing the command, but to no avail. What am I doing wrong here?

You need to add your current user to the docker group as follows:

sudo usermod -aG docker $(whoami)

then logout & login again into the system or restart the system. test by docker version

for further info how to install docker-engine follow docker documentation