Hey, I have a DM5 with Cubase essential 5 so that looks familiar 
There’s 2 sides to recording electronic drumkits.
- audio
Recording audio means you capture the actual sound that the DM10 generates. This requires you to connect the audio outputs on the DM10 to the inputs on your audio interface. To record in Cubase, create a stereo audio track, set it’s input to the audio interface inputs you just connected your DM10 to, it’s outputs you can route anywhere you want, default is to master output which is fine for regular playback.
The monitor button which Nate mentioned you will find on every track, it switches the input of said track from what is already recorded to what it hears at the input. Generally you turn it on when recording and then off to listen back the recording.
If everything works so far, you should see a waveform from your recording in cubase. If not, something is wrong with the wiring or internal connections. Check the VST connetions window (in one of the dropdown menus) to see if your audio interface has the right in and outputs selected and available. You should also make sure you have the correct driver for your audio interface running, you can change that in another window I forgot the name off 
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–edit–: it’s device setup, look for the VST tab, that’s where you can change drivers. Generic ASIO and DirectX may be your only options, both are useless but try both to see which works best. You can also download ASIO4ALL drivers and try those instead. Typical problems with onboard soundcards include crackles, huge delay in playback etc.
–/edit–
If you’re using onboard sound you will be out of luck, these usually don’t perform well because their drivers are not made for this type of application. It’ll probably work but not as fluent as you’d like.
Report back where things go wrong if you can’t figure this one out 
2. midi
The wonderful advantage of using an edrum kit is Midi recording. Midi is not sound, it’s what I call ‘instructions’ on how you play the instrument. These instructions you can record and then play back to the DM10, and it won’t notice a difference between you playing it directly or you sending midi to it. The advantage is 2fold:
- you can use your drumkit to trigger other sounds than just those in the DM10 module, like for instance EZdrummer which you mentioned, or other soft-or hardware instruments.
- you can make corrections to your playing and record the audio from the DM10 after you made corrections.
To record midi: You need 2 midi cables (5-pole DIN cables which are very cheap.) and a midi interface. For just recording midi you can do with 1 cable, but if you want to playback the just recorded midi to your DM10 again you’ll need 2. Maybe the DM10 has a USB connection, in which case you don’t need all this.
Wire Midi out from the DM10 to Midi in on your interface and vice-versa. In cubase, create a midi track and select the desired inputs and outputs. When playing the DM10, you should see a red bar indicate the reception of midi data in Cubase, next to the audio meters on the transport bar (f2).
You should now be set for recording midi. If you plan on using the recorded data for drum triggering (external or sofware), I suggest you set the drummap of the miditrack to GM. (this is below the midi input and output section of the midi track.) This will open the recorded data in the drumeditor with most of the pitches assigned to the right sounds right away. At least on the DM5 the GM map is mostly accurate. You can always make your own mapping later if you wish, but that’s not important now. If you played to the metronome, you can now use the Grid in cubase to allign hits that were off, or use quantise to correct a whole section if necessary.
There’s a lot more you can do now, but again: report where this goes wrong and we’ll go from there
–edit-- I’m curious what your setup is like. When playing the DM10, how do you monitor yourself? Do you have speakers connected to it? And when recording, do you rewire the DM10 to your soundcard instead?
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