By efficiency, we are really saying, what percentage of the power that goes into the inverter comes out as usable AC current (nothing is ever 100% efficient, there will always be some losses in the system). This efficiency figure will vary according to how much power is being used at the time, with the efficiency generally being greater when more power is used.
Efficiency may vary from something just over 50% when a trickle of power is being used, to something over 90% when the output is approaching the inverters rated output. An inverter will use some power from your batteries even when you are not drawing any AC power from it. This results in the low efficiencies at low power levels.
A 3Kw inverter may typically draw around 20 watts from your batteries when no AC current is being used. It would then follow that if you are using 20 watts of AC power, the inverter will be drawing 40 watts from the batteries and the efficiency will only be 50%.
A small 200W inverter may on the other hand only draw 25 watts from the battery to give an AC outpur of 20 watts, resulting in an efficiency of 80%.
Larger inverters will generally have a facility that could be named a "Sleep Mode" to increase overall efficiency. This involves a sensor within the inverter sensing if AC power is required. If not, it will effectively switch the inverter off, continuing to sense if power is required. This can usually be adjusted to ensure that simply switching a small light on is sufficient to "turn the inverter on".
This does of course mean that appliances cannot be left in "stand-by" mode, and it may be found that some appliances with timers (eg washing machine) reach a point in their cycle where they do not draw enough power to keep the inverter "switched on", unless something else, eg a light, is on at the same time.
Another important factor involves the wave form and inductive loads (ie an appliance where an electrical coil is involved, which will include anything with a motor). Any waveform that is not a true sine wave (ie is a square, or modified square wave) will be less efficient when powering inductive loads - the appliance may use 20% more power than it would if using a pure sine wave. Together with reducing efficiency, this extra power usage may damage, or shorten the life of the appliance, due to overheating.