Thanks for taking the time to ask your question Luke. A half-Senate election is when only 40 of the 76 Senate seats are available for election (half the state Senate seats plus the 4 senators representing the Australian Capital Territory and Northern Territory). Half-Senate elections usually take place at the same time as House of Representations elections (ie at least every 3 years). More information about federal elections can be found here.
In comparison, a full Senate election (all 76 seats) only occurs after both the Senate and the House of Representatives have been dissolved in a double dissolution.