To verify means to check information to see if it's true or real, usually by checking with an independent source.
Ex: Ms. Snyder seems like a great candidate because of her impressive CV. But have you called her former employers to verify her previous work experience yet? We must do this to assure the Human Resources manager that she is the best candidate.
Ex: Yes, Derrick was definitely ill last night. I called his doctor to verify it; he was even in the hospital for a few hours.
You can use the noun form, verification, to describe the process of checking the truth or accuracy of something.
Ex: Sally impressed the managers at her interview. Now they've requested her list of referees, so they can begin verification of her previous experience. This will mean calling her previous employers and colleagues to ask about her experience specifically.
Ex: When doing online banking on our website, customers must enter their personal identification number, their birth date, and their postal code for verification. This way we know that only customers are using the system, because a third party will not likely know these details.
To claim means to say something is true, usually without evidence or proof.
Ex: Sarah claims to be a good cook, but I've never tasted her food, so I can't say from my own experience if it's true or not.
Ex: Dan has always claimed to be an excellent singer, but we'll be able to verify that tonight at the karaoke party.
To claim can also mean to say that something is yours. With this meaning, we often use the phrases to stake a claim to something or to lay claim to something.
Ex: When we arrived at the hotel, Sarah entered the room first. She immediately claimed the bed next to the window.
Ex: Since Sarah laid claim to the bed next to the window, I had to take the other one. But I will stake a claim to the window seat on the train, so I can look out as we ride.