When it comes to boilers, steam units are clearly outdated, and electric boilers have very high running costs. Natural gas boilers are now the standard in hydronic (hot water) heating; oil and pellet boilers have a much smaller market share.
This page deals with the differences between these types of boilers.
Just do not forget that the energy efficiency of any boiler depends largely on a wide set of factors besides their fuel: proper installation, sizing, servicing and AFUE (efficiency coefficient)... And that hydronic systems may not make sense in low-energy homes.
Steam boilers are very different from water boilers; they do not rely on pumps to move the water in the pipes; they boil water to create steam, which then expands and fills the pipes and radiators, distributing heating through the house.
That’s a simple system, typically reliable, but not efficient. It's very difficult to control temperatures with steam boilers.
If you have a steam-radiator system you may be able to retrofit it for water distribution. Water boilers are more efficient, and the upgrade will provide better comfort, reduce the noise and create the possibility of zoning (room-by-room heating). Switching fuels is rarely cost-effective, but in the case of steam boilers it may be worthwhile.
Like electric furnaces and other types of space heaters based on electric-resistance elements, electric boilers have very high running costs.
Only heat pumps may provide cost-competitive central electric heating, but they only perform well in climates without significant periods of freezing temperatures.
Gas boilers have much lower running costs than electricity, and new units provide a clean and safe combustion, namely gas condensing boilers (the most sophisticated, expensive and high-performance type of boilers).
Wall-hung boilers are very popular in Europe. They are relatively cheap, and provide a very precise control of gas flow and combustion air. Wall-hung boilers are typically air-sealed units, and law requires them to be of condensing type, that is, they are able to re-use the water vapor and gases they are producing.
These boilers are different from common North-american boilers, for single and multifamily homes: the sectional boilers, the steel fire-tube boilers and the copper-water tube boilers. American boilers are typically intended to supply larger amounts of hot water, and are on average a lot more expensive.
Oil boilers are the most common alternative to gas boilers in regions without access to the natural gas network. They aren't as clean, as safe or as efficient as gas boilers can be, but modern units are quite sophisticated.
Top oil boilers have AFUEs (energy efficiency coefficients) of 80% to 97%, which means that they are able to provide energy savings of at least 20% when compared to older units, with AFUEs of less than 70%...
Though there are now condensing oil boilers, they have a small market share, partially due to technological flaws: small amounts of gases/water vapor to reuse, and problems of corrosion in the flue and heat exchanger.
Anyway, their efficiency depends largely on proper installation, sizing and servicing (oil boilers should be professionally serviced yearly).
Modern pellet boilers can be sophisticated and expensive machines (above $7.000/$8.000, for large capacity boilers). State rebates and incentives can make them more competitive, but they only make sense for large heating needs in buildings with low levels of insulation and air sealing. That is: they do not make sense in energy-efficient single-family homes.
These machines use microprocessors to control the amount of fuel and combustion air, and their feed system is based on an automatic suction mechanism or on an electrically spiral conveyor. They are low-maintenance units: even ash removing is now largely automatic; some units can automatically compress the ash in a storage container, and you just have to remove it once every two or three months.
Like modern gas condensing boilers, these pellet wood boilers are sealed-combustion units (direct-vented through PVC pipes...). This feature solves the issue of the acidic exhaust gas in old and unlined chimneys.