Every solicitor or solicitor’s firm (but not barristers) has two accounts: a client account and an office account.
The office account is the firm’s own money to spend as it wants: eg payment of wages and other expenses (rent, electricity etc) and to the solicitors themselves.
The client account represents a lot of relatively much smaller accounts to which each individual client pays money: for example, a solicitor may ask for money on account from a client before the solicitor does work, or continues with work, or where a solicitor wants money to pay to eg barristers or other fees or expenses (which solicitors call ‘disbursements’). That money must stay in ‘client’ until the solicitor (1) uses it for expenses or (2) sends the client a bill. Once a bill is sent the solicitor can take money from ‘client’ to ‘office’ and it becomes the solicitors’ own money. Any balance on ‘client’ remains money which belongs to the client.
Money on account is a term used by solicitors for them to receive money from a client to be paid into their client account: (1) so the solicitors know they are not doing a lot of work without any guarantee of payment (they cannot pay it to ‘office’ because they have not yet done any work to render a bill); and (2) to give the solicitors a fund from which they can pay disbursements.