How do I eliminate them??

Well there are a few ways to be sure they don't happen, or at least don't happen regularly.
Insulate and Ventilate
Ice belts
Heat tracing

Insulate and ventilate the attic.

You want a cold roof.. that will keep the snow from melting on the roof. At least when it does melt on the roof, it will not freeze again when it reaches the eaves. Ideally what you want is a roof that is a constant temperature from top to bottom, from eaves to peak.

Insulate
So, adding insulation to your attic floor will help. You should have enough insulation in your attic floor to give you R-49 (or you don't live in an area where ice-dams are a problem). That is about 20 inches of fiberglass. It means giving up the attic floor as a place to store things, since to get that much insulation you will have to go deeper than the joists, and preferrably lay it perpindicular to the joists. You should NOT use insulation with a vapor barrier (kraft faced) since you should not add a vapor barrier part way through your insulation. Before laying the insulation down, or before blowing it in, seal up any openings in the floor. Electrical wiring comes through holes, caulk them closed. The plumbing vent lines runn through the attic... caulk around those pipe openings. If you have light fixtures which require the insulation to be left away from them to dissipate the heat, then make sure you do that... you don't need to create a fire hazard here.

At the eaves, make sure the insulation does NOT close off the opening between the rafters that allow the air to flow up from the eave (soffit) vents. If you are blowing in insulation or packing the batts in tight there, buy prop vents. These are styrofoam panels that fit between the rafters to keep a channel open.

Ok.. you are stopping the heat from escaping into the attic.. That is good just to save on your heating bill as well as help to eliminate ice dams. Now lets talk about...ventilation

Ventilation.
Ventilation is important for many reasons. Key among them is the elimination of moisture that finds its way up to the attic. (This is the reason for sealing the holes) Moisture that makes its way to your attic will condense in the cold air. It will condense on your plywood roof decking and cause it to rot, and it will pass up through the shingles causing them to fail years before they should.

It also keeps the attic cool in summer, which is also good for the roof, and good for your comfort in summer. Yes you have all that insulation up there, but an attic that is 160 degrees will certainly make the rooms below the attic very warm indeed.

Finally.. for ice dams, a well ventilated attic will be colder.. that means less snow melting on the roof. The closer the temperature of the roof is to the temperature of the part over the eaves, the less melt you will get when it will freeze again on the eaves.

How much ventilation should you have. If you have ice dams, then the recommended MINIMUM is 1 sq ft for every 150 sq. feet of attic floor area. Adding more will only help. Ideally you want it split 50-50 between the soffits and the peak. Best is a continuous soffit vent and a ridge vent.
Again, make sure the paths from the soffit vents up between the rafters is free of insulation so the cool air can rise up to the ridge of gable end vents.

By the way, mechanical ventilation is NOT recommended. If a fan pulls a negative pressure on the attic and the negative pulls moisture into the attic from the house, it can create a new problem of condensing moisture in the attic.

Flashing

Ok, either you have alot of insulation and ventilation and you still have ice dams, or, you have cathedral ceilings or a roof attic design that makes adding those impossible. In that case you can have an ice belt installed.
An Ice Belt is Metal flashing installed all the way along the eaves and extends to the area of the roof above the attic. It covers the area where the ice will form. It serves two purposes, first it provides a surface which permits the snow and ice to just slide off preventing most dams. Second it provides a surface that water won't flow up under as it as it does with shingles.

There are other membranes that serve the same purpose to keep the water out, that can be installed under the shingles at the eaves. Again, extending up the roof, it will keep the rain that gets under the shingles from wetting the roof deck and leaking into the house.

Removing the rain gutters is often necessary since they can become a blockage holding the snow and ice at the edge of the roof causing a dam.

And you can always install a metal roof. These, while they are expensive, will eliminate ice dams completely. The snow and ice will slide right off, and they have no horizontal seam that water can get up under.

Heat Tracing

There is electricity. Heat tracing can be run along the eaves that you can turn on should an an ice dam begin to form which will melt the snow and ice on the eaves, eliminating the ice dam. This is relatively easy to install and relatively inexpensive to purchase. It only needs to be run when a dam is forming, so it is not a large energy user.

Finally,

You can buy a roof snow rake and rake the snow and ice off the edges of the roof as soon as it forms. Ok, this method may be the cheapest.. but.. for very high roofs... yikes, pulling all that snow down on yourself!
However, if you have an ice dam right now.. this winter.. and water is coming into the house, then this may be your best bet right now. Getting the snow and ice off the roof will stop the source of water leaking into your house. Continue to keep the snow off the roof until you solve the problem with a long term solution. If you have to climb up there to do it.. BE CAREFUL!