The phrase that we use to describe putting money into a bank account is "deposit". You can follow the word "deposit" with a few different things:

• deposit (some cash / a check / a money order): "Can I deposit this check please?"
• deposit (an amount): "I deposited two thousand dollars into your account last week."

You deposit something into an account. In the example above, the speaker says "deposit this into checking". It's understood that this means "into my checking account".

A checking account is a bank account that you can easily get money in and out of with checks (small pieces of paper that you fill out and sign to pay someone with) or a debit card (a card that you pay with which takes money directly out of your bank account). Checking accounts usually don't earn interest. That means that you don't earn extra money for the money that's in a checking account. Other accounts, like a savings account, do earn interest.