There has been a lot of talk about power sources and their efficiency. I'm just wondering about how effective a thermoelectric generator's efficiency compares to mechanical turbines, solar panels, etc.
Efficiency may not be the appropriate metric, if you are interested in comparing large-scale power sources.
The efficiency of a solar panel is the ratio between the electrical output and the incident solar energy. If we know the average characteristics of sunlight in a given area, the efficiency allows us to calculate the output power for a given panel area (or vice-versa). The efficiency allows a useful comparison between two solar panels, but doesn't help us make a comparison between e.g. a solar installation and a wind farm.
The closest equivalent figure for a turbine installation might be the generator efficiency - the ratio between the electrical output and the mechanical energy extracted by the turbine. But the mechanical energy extracted by the turbine is only a teeny tiny portion of the total mechanical energy that could theoretically be extracted if the turbine was unconstrained by physical or practical considerations. In this sense, a wind farm's efficiency is very low. But generating power from the wind turns out to be economical nonetheless. We don't have to extract a large percentage of the available energy to make it worth it.
Thermal generators are similar in this respect - they are not efficient in the sense that they make use of a minuscule percentage of the power that could be theoretically be extracted from a given thermal gradient. But that doesn't tell us much about whether they could be useful for meeting our power needs, or how they compare to other power sources. That sort of question would require a much more in-depth analysis that accounts for cost, environmental effects, etc.
To understand the potentials and limitations of thermoelectric generation, you would need to start by identifying the thermal gradient serving as the source. Is it geothermal? Is it waste heat from another process? Once you have a source, you can start to quantify the available energy. Only then does the efficiency come into play, along with cost, maintenance requirements, portability, reliability, and so on.
I get the feeling your question was in the context of grid power. To my knowledge, TEGs are not a major competitor to e.g. solar or wind power in this arena. TEGs are used to extract energy from very high thermal gradients, of which few are renewable and naturally occurring. Geothermal gradients are exploited in some cases as a renewable source of substantial power, but TEGs are not the best technology to extract this energy because the gradients occur over a large distance.
Of course, a TEG could be used to generate electrical energy from a thermal gradient generated by burning a fuel - this is actually done in some cases. But here, the very low efficiency of TEGs makes them uncompetitive with mechanical generators on a large scale.
There is at least one "naturally occurring" thermal gradient that could be exploited. A burner (say of wood pellets for the sake of renewable energy) that is required to heat a home with a radiant floor at less then 110 F and a heating load of continuous burn of about 17,000 BTUs/hr for many months in winter. That would burn about 2 pounds of pellets per hour. At night the household use of electricity is low and not available from solar, but a pellet burner burning close to 24-7 could supply 300~400W day and night from a 15,000 to 18,000 BTU/hr heat source (something over 5kW equivalent). That would be more than enough electricity to power the burner and pumps for the system. To keep TEG efficiency optimum the combustion chamber surface would need to be hot (about 500F or hotter but below the TEG melt threshold) at the surfaces where TEGs are heated. The combustion gas at that temperature is too hot to discharge unused, so the exhaust gasses would need to be cooled to recover heat that would otherwise be lost. Fresh air for combustion could be preheated several hundred degrees with the hottest exhaust as it exits the area of TEG heating. That would improve the combustion efficiency. Finally a water jacket on the final exhaust discharge could be used to preheat domestic hot water storage (reverse flow from 50 or 60 deg F water input to as hot as it happens to get going back to the water tank). The tank would need to be a pre-heat tank and would need to be large to prevent overheating the system when hot water is not being used. The heat from the burner would be passed through the TEG to the radiant floor as the "primary use" is for the space heat, and 6 or 8% would be converted to electricity. So the system could be extremely efficient even if the TEG itself is not. A very large percentage of the total combustion heat would be recovered and an adequate amount of electricity would be produced. This co-generation is a natural fit. If the system's electrical output were applied to a battery off-grid system it would contribute substantially to the total electrical load generating about 8 kWh per day, however consuming about half of it for it's own operation. It would probably supply almost all of the domestic hot water heat as well as heat the house, and would still keep the batteries charged at night or on cloudy days when PV is not available. Sorry, it will take a long time to charge the Tesla.