Often parties to a negotiation will make a certain amount of progress, then get stalled. Each side may have made what they feel are reasonable compromises in their positions, but have arrived at a point that is still distant from the other side's position. Mediators use various techniques to bridge this gap, which may be as simple as calling a break, or may require getting the parties to consider a mediator's proposal. I see it as a process of getting both sides to cross a line they did not want to cross before the mediation, and often the way to make them do that is to make them understand that the other side is making a similar leap of faith.
To break an impasse, it might also be helpful to consider what got the parties into the impasse in the first place. As an example, let's look at the biggest impasse that is facing the people of California right now: the stalemate in the state legislature between Democrats and Republicans over how to deal with a revenue shortfall of more than $20 billion. Although the Democrats control the legislature, there is a two-thirds requirement for raising taxes, giving the anti-tax Republicans approximate parity in power. The Democrats refuse to cut spending sufficiently to bridge the gap; the Republicans refuse to raise taxes. So the state has reached paralysis, and has even resorted to issuing IOUs instead of cash, which it does not have. To a mediator, the obvious solution is some combination of tax increases and spending cuts, and the question is what is keeping the parties from considering any sort of reasonable compromise solution.
One possibility is that each side actually believes its own rhetoric. The Democrats believe that drastic spending cuts would create a worse situation for the state than being unable to pay its bills; and the Republicans similarly believe that it would better for the state to fall off a cliff than to raise taxes by another penny. Of course, both sides can't be right, and they're probably both wrong, but it's difficult to talk them out of their positions. What they need to do is listen to each other more, which might make some people at least wonder a bit about whether all of the truth and right are on their side.
A second possibility is that the parties are so hung up on the past that they cannot address the future. This frequently happens to parties to a lawsuit as well. They are involved in a process that seems to reward them for making charges and counter-charges, for assessing blame and seeking penalties for past misdeeds They are trying to win the blame game, and are not interested in playing a different game, called solving the problem. I have been arguing about the California budget crisis on my political blog, one of the joys of blogging being the opportunity to trade jabs with people of differing views, and when I suggest that the state needs to solve this problem with a combination of tax increases and budget cuts, I am met with the counter-argument that it is all the Democrats' (and perhaps the governor's) fault for letting spending get out of control in the first place. The logic being that if it is all one side's fault, that side should pay. Why should the other side have to sacrifice its principles so as to solve the problem? You can never totally get past the blame game in a mediation, because considerations of fault must enter into the parties' assessments of a fair outcome, but sometimes you do have to point out that the only way to resolve the issue of fault is through a different process that carries its own substantial costs and risks, and doesn't always necessarily arrive at the result that the parties are expecting. The negotiated resolution has to move beyond the fault-finding stage.
A third and perhaps the most important reason for impasse is based on calculating the political pressures on each party. Right now both parties in the state legislature perceive some political gain from holding fast to their respective positions, and both would also face substantial political costs in surrendering their positions. Any Republican who votes for a tax increase is likely to be defeated in the next primary by a more strident anti-taxer. Similarly, Democrats who vote to allow drastic cuts in programs that benefit their constituents will face the wrath of the voters also. This dynamic is not going to change until these politicians perceive the political costs of failing to reach agreement as greater than the political costs of making an agreement. These kinds of pressures can also come into play in a private negotiation. Parties often have to explain a settlement to the powers that be, whether that is a spouse or the board of directors. Sometimes it is easier to keep the dispute alive and fight for your position, even in the face of mounting costs and possible defeat, than it is to explain how and why you gave in to make peace. As long as you keep fighting, you can continue to hold onto your own conception of the worth of your position. Only after you either settle or allow a neutral party (the judge for litigants, or the voters for politicians) to decide your case, do you have to face the reality of what your case is actually worth.
(photo from San Francisco Sentinel)