As a dog sitter, you do not necessarily need to be bonded unless your employer requests this or you manage other employees as part of a pet-sitting business. As a house sitter, you would need a fidelity bond if the people whose home you're watching ask for one.
Fidelity bonds basically protect your employer, or client, from theft, according to Sittercity.com, which has a section for pet sitters. You can think of these bonds as being like insurance policies.
If there is a theft during your work at a client's home and you are found guilty but have established a bond, "the company who holds [the bond] will make an immediate payment to your client, up to the value of the bond," according to petsitterportal.com.
According to Sittercity.com, you will want to be bonded if you run a pet-sitting business: "Typically, only pet-sitting businesses with more than one employee will need bonding, since it protects against employee theft."
If you are a pet sitter with no employees, it's not necessary to be bonded, according to Sittercity.com. If you house sit, however, your client may ask that you be bonded.
You may wish to become bonded as either a pet or house sitter of your own accord, however, because, as Sittercity.com says, it's "a good marketing tool" as it gives clients a sense of protection.