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 four 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 three years). More information can be found here.
In comparison, a full Senate election (all 76 seats) only occurs after both the House of Representatives and the Senate have been dissolved in a double dissolution.