A conversion date is the date that you start using a new accounting software.
A conversion balance is the amount for a particular account as at the conversion date.

Say you started using Xero on 1 July 2015.
Your conversion date would be July 2015. All invoices on or after that date would be entered into the system. However, you might have traded prior to that and have amounts in your bank account and hold stock. These would be conversion balances because you need your accounting system to reflect what you own.

So your conversion balance, as an example, would be:

Bank $4000
Stock $500

If you enter conversion balances in Xero, you’ll have reports that show you your ‘true’ financial picture because it will include the amounts you’ve made and value of things your business owns prior to going onto Xero.

The reason why conversion dates and balances are important is because things can get messy with accounts receivable/payable and all the transactions that occur that a conversion date marks a line that clearly differentiates from the transactions that occur on and after conversion date to the ‘historical’ stuff which needs to be in the balance.