Savings Account
This offers you a basic banking service with which you can make purchases and withdrawals. Usually you can also access your savings account using the bank's smartphone app or website. You will typically get a debit card linked to the savings account which you can use at ATMs and in shops. I don't think you can use these debit cards (backed by VISA Electron or Mastercard Maestro) for internet shopping, but FNB offers a VISA card on their Smart savings account, which is a cross between a credit card and debit card. It works like a credit card in that you can use it for internet shopping, but like a debit card in that your purchase transactions are deducted from your savings account and the card offers no credit facility.
The name "savings account" is a misnomer. It pays very low, and in some cases zero, interest. The interest rate is usually lower than the inflation rate, so money is actually losing value in a savings account.
Cheque Account
I haven't written a cheque in years, not since the advent of electronic banking, but the distinguishing feature of a cheque account is that it comes with a cheque book. Also, usually, in order to get a credit card, you would need to open a cheque account first. Like credit cards, cheque accounts also usually provide a credit or overdraft facility. And the banks like to provide a whole host of add-ons to cheque accounts, such as travel insurance and garage cards. The latter works like a debit card, but only for fuel. These days you can also use a credit card for fuel purchases.
Because of the credit facility that comes with a cheque account and a credit card, these accounts are required (I think) to build a credit record.
Usually you earn zero or close to zero interest on a cheque account.
Transmission Accounts
This is an old type of account. I doubt that any of the banks still offer new transmission accounts. Practically it functions very similar to a savings account, and is essentially a transacting account, i.e. an account to pay money into, withdraw money from, etc.