A settlement is legal terminology meaning that a matter before the court has been resolved.
When two opposing parties to a civil lawsuit are able to achieve such a resolution before the judge or jury render a decision, it is called an out of court settlement.
Because court trials are expensive and time consuming, both parties in any civil case will do their best to reach an out of court settlement whenever possible - preferably before the start of proceedings. Out of court settlements may be discussed in the offices of one party's counsel or the other, although it is usually the defendant that proposes the settlement.
A defendant may offer to settle at any point prior to the judge's decision, and will do this for any number of reasons. Usually, it is to avoid the time and expense involved in litigation. In a high-profile case involving a well-known corporation, there may be an additional incentive to avoid publicity that would reflect badly on the company's image (although this is becoming increasingly difficult for corporations to overcome).
Either party may offer to settle before the judge or jury reaches a decision. When a plaintiff offers to settle, it is usually because the attorney feels that the case is not going their way; there is a possibility the judge or jury may find in favor for the defendant. Of course, a defendant may also offer to settle for a similar reason.
An out of court settlement offers is usually smaller than the one being sought by the plaintiff. When a defendant makes an offer and the plaintiff accepts it, the legal proceedings stop; the plaintiff surrenders any further claim on the defendant and forfeits the right to pursue further legal action on that particular issue. Such a settlement may have other conditions attached as well.
Once a plaintiff accepts an offer to settle, the defendant is under no obligation to admit liability or fault.
The closest analogue to an out of court settlement in criminal court would be a plea bargain.