You have two options:
- use tandem breakers
- Install a sub panel
There are caveats with both routes. When using tandem breakers on a 120 volt system (i.e. with a neutral present), you want to avoid something that is called a multifeed. This is, two circuits on the same phase sharing the same neutral. When using tandem breakers, its very easy to inadvertently do this.
If you are in the US, you will see two feeders from the meter, one of them probably has some red tape on it, the other is black. These are your phases. Normally, every other breaker is on a different phase, but tandem breakers put both circuits on the same. Take care that both circuits attached to a tandem have their own neutral (white wire).
While you might be able to install a tandem breaker yourself (I highly recommend calling a qualified professional), you will surely want an electrician to install a sub panel. They aren't much different from a regular panel except:
- Ground and neutral conductors are isolated, not bonded on sub panels
- You want to watch your loads
- You'll want to be careful about where you place the breaker that feeds a sub panel, so that you don't develop a hot spot on the main panel bus. For instance, you don't want a 100 amp sub panel breaker right in the middle of a water heater and an air conditioner.
In either case, I really recommend calling an electrician.
Without being able to see the cables as they enter the cabinet; or the ability to touch or trace them, here is what I assume is going on.
Definitions:
Grounded (neutral) from the service
A typical single split phase service is made up of 3 wires. Two ungrounded (hot) conductors, and one grounded (neutral) conductor. The ungrounded (hot) conductors will connect to the main service panel through a disconnect (usually a large breaker), while the grounded (neutral) connects to the neutral lug. The neutral lug will be bonded (electrically connected) to the neutral bus bar, and all grounded (neutral) branch circuit conductors will terminate at the neutral bus.
Grounding Electrode Conductor
This conductor is used to connect the grounding electrode (ground rod, etc.), to the grounding bus in the panel. All equipment grounding conductors will be connected to this bus.
Bonding Jumper
The bonding jumper is used to bond (electrically connect), the un-energized metal parts of the panel to the grounding system.
Assumption:
Since it appears that (what I assume is) the grounding electrode conductor terminates at the neutral bus, I'm also assuming that this is the main service disconnect. This leads me to believe that the neutral and grounding buses are bonded (electrically connected). In which case, technically, grounded (neutral) branch circuit conductors can terminate at the grounding bus.
So you have two options:
Terminate the grounded (neutral) from the new circuit to the grounding bus.
Move the green wire that is terminated on the neutral bus, to the grounding bus. Then terminate the grounded (neutral) from the new circuit, to the freed up slot on the neutral bus.
Additional Information and Code Compliance:
Number of Conductors
Since this is a new circuit, it has to be installed to current code standards.
National Electrical Code 2011
ARTICLE 250 — GROUNDING AND BONDING
VI. Equipment Grounding and Equipment Grounding Conductors
250.140 Frames of Ranges and Clothes Dryers. Frames of electric ranges, wall-mounted ovens, counter-mounted cooking units, clothes dryers, and outlet or junction boxes that are part of the circuit for these appliances shall be connected to the equipment grounding conductor in the manner specified by 250.134 or 250.138.
Which in this case means installing a NEMA 14 receptacle for the dryer, and a proper grounding conductor.
You'll have to follow the dryer manufacturers installation instructions for upgrading to a 4 wire cord. For more information see this answer, and this answer.
Since you've said that you're already using 4 wire cable, you'll simply have to terminate the grounding conductor in the cable to the grounding bus in the service panel. Then connect the other end of the grounding conductor to the grounding terminal in the dryer receptacle.
Size of Conductors
You'll also want to be sure that you're using the proper size breaker and conductors. In the case of a dryer, you'll typically use a 30 ampere breaker and 10 AWG conductors (depending on the length of the run). However, you'll want to check the dryer manufacturers installation instructions to verify this.
Best Answer
Read my answer here that talks at length about what "spaces" are and how duplex breakers work.
It appears that you have one space left. Any 2-pole (240V) breaker needs to sit in 2 spaces so something special will have to be done. The article I linked should give you plenty of ideas, e.g. using a duplex breaker for two 120V circuits to free up a space.
If you expect to add any more circuits in the future, you may be better off just adding a sub-panel.
Lastly while you're in the panel you might look at a few minor problems. First, some of the white wires going into breakers are not neutrals. They should be taped or marked at both ends to indicate they are not actually neutrals. Second, it looks like too much bare wire is exposed on most of the wires going into the breakers, probably too much insulation was stripped (there's a "how much to strip" guide printed or embossed on the breaker.) It can be easily trimmed back.