There are different kinds of bonds that can be used in business situations. Examples include performance bonds, bid bonds, court bonds, and fidelity bonds. (The Wikipedia page on surety bonds lists many more examples, with links to the more common ones.) All involve a third party guaranteeing an obligation of some sort–if the bonded person or business does not complete the job, for example, the funds supplied by the bond will ensure that someone can be hired to do so. Or, if the bonded person has charge of a business’s funds, and steals them somehow, a fidelity bond makes sure that the business is paid back.
In this sense, bonds are like insurance, though they are not strictly insurance. Insurance is a two-party contract (insurer and policyholder); bonding involves three parties (principal, obligee, and the bond issuer; also often know as the “surety”). Given this, I am doubtful that your average householder plumbing customer (the principal) would have taken the time to get the bond issuer (the surety) to write a bond for a plumber (the obligee) in order to make sure the job is completed. However, the householder would be glad to know that the plumber has gone to the trouble of buying a policy of liability insurance–this helps protect the householder from the plumber’s actions.
But the same plumber also could be hired as a subtrade to a general contractor doing a much larger job (say, plumbing an office building under construction). In these situations, it is important that the job be completed by a certain date so other subtrades can begin work (for example, plumbing must be done before the drywallers go in). In this case the general contractor must be assured that a subtrade will finish on time, and so, might require that that subtrades either be bonded, or be bondable. Note that if the subtrade, through a delay or screwup, causes the surety to have to make good on the bond; the surety has a right of subrogation against the subtrade–it can use all legal methods to get the subtrade to pay back the funds the surety had to expend on its behalf. So it is in the bonded subtrade’s best interest to make sure the job is done right and on time.
At any rate, you should be able to see that the plumber–or the electrician or carpenter–could offer his or her services to a wider market by advertising that he or she is “bonded [or bondable] and insured.”
The only other thing about “bonded and insured” that you should be aware of involves fidelity bonds, and this is what cher3 touched upon. At one time, people who dealt with cash or negotiables were always covered by fidelity bonds; and while fidelity bonds are still available, it is much more common today to buy fidelity insurance to protect against employee theft or embezzlement. But because the two products are still offered, a person such as a bank teller could be both “bonded and insured.”
This is a very general overview of a complex topic–you can, if you like, take entire courses on bonding; although I will admit my knowledge only comes from reading a bonding course textbook I used as part of some research I did a few years ago. But I hope it helps somehow.