One can identify energy with mass via Einstein's oft-quoted equation E = M.c2 although I would generally prefer to express this as E/c = M.c, since this is the value of the time-like component of the 4-momentum of a particle. Since the factor of c2 is close to 9e16, ninety thousand million million, the energy equivalent to a mass is described by a bigger number, in SI, the Système Internationale, (and even more so in more archaic systems). Consequently, I list some of the tinier masses here under their energy equivalents and some of the huger energies under their mass equivalents. The SI unit of energy is named after James Joule. Nuclear physicists also routinely use the electron Volt, eV, as a unit of energy: this is the amount of energy that an electron gains or loses when moving between two locations whose electrostatic potentials differ by one Volt, all other things being equal: it is very small.
One eV (electron Volt) is 0.160210 aJ; its mass-equivalent is 1.78e-33 g. The time-scale on which the uncertainty principle obliges even the law of conservation of energy to tolerate discrepancies of eV order is around 4 femto-seconds; this is the time light takes to travel 1.24 microns. Last time I checked (pre-2008), the mass of neutrinos was known to be at most 46 eV, or 82e-33 g, equivalent to 7.3697 aJ. The Rydberg energy, 13.6 eV, is 2.18 aJ; this is the energy needed to ionize an atom of Hydrogen in its ground state. Doubling this last, we get the Hartree energy, 4.358 aJ; scaling this by the square of atomic number serves as a rough estimate for the total energy needed to strip a full shell of electrons from an atom. I thus estimate that it takes about 17 aJ to totally ionize one 2He, helium, atom (which has exactly one full shell of electrons).
Continuing, after He, with the total ionization energies of the noble gasses (which have only full shells, each having one more than the previous) I estimate: 10Ne (with two shells) at 0.87 fJ, 18Ar at 4.24 fJ, 36Kr at 22.6 aJ, 54Xe at 64 fJ, 86Rn at 193 fJ (which is a little more than the energy an electron and positron release upon mutual annihilation) and element 118 (of which only three atoms have been seen) at 425 fJ (or 2.65 MeV). The energy-equivalent of an electron's mass is 81.87 fJ, just over half a MeV; uncertainty admits discrepancies of this order for time-scales of order 8 zepto-seconds, during which light travels of order 2.4 pm, which falls between the scales of nuclei and atoms.
The electrons in Earth's outer van Allen belt have energies up to 10 MeV or 1.6 pJ; protons whizzing around in the inner belt have energies up to about 400 MeV or 64 pJ (these can get through a seventh of a metre of lead; that's almost half a foot). A significant flux of antiprotons has also been observed in the van Allen belts, with energies in the 60 to 750 MeV range, from 10 to 120 pJ (a significant fractionof the energy equivalent of their mass).
The energy equivalent of the atomic mass unit, AMU, is just under 0.15 nJ; the energy equivalents of the masses of a free proton and a free neutron are just slightly over 0.15 nJ. The energy-equivalent of the mass of one 12C carbon atom is 1.79 nJ or 11.18 GeV.
The LHC, CERN's Large Hadron Collider, can get individual protons close enough to the speed of light that their total energy is 7 TeV (less than a millionth of the nutritional energy content of a milligram of sugar, but still a huge amount by the standards of nuclear physics); that's fractionally more than 1 µJ and they'll be crashing pairs of such into one another, yielding twice as much.
The LHC can accellerate ions of 82Pb, lead, to similar speeds, giving energies 82 times higher; when a pair of these collide, the total energy is 1148 TeV, or about 0.18 mJ.
The mass-equivalent of 1 J is 11.1265006 femto-grammes. When I roll over in bed, of order 100 kg moves up and down by of order a tenth of a metre in Earth's surface gravity, expending of order 100 J.
One gramme of trinitrotoluene (TNT), when detonated, releases about 4.184 kJ of chemical energy; this is the energy-equivalent of 46.55 pico-grammes. The British Thermal Unit is 1.0547 kJ. Lifting me through a metre in Earth's surface gravity costs about one kJ; when I climb the stairs to my flat, I expend of order 10 kJ.
One kg of TNT, when detonated, releases about 4.184 MJ of chemical energy; this is about 1.16 kWh (i.e. kilo-Watt hours). On a walk or cycle-ride from central Oslo up to Maridalsvannet, a pleasant afternoon's tour, I ascend about half a kilometre; aside from the effort of moving around generally, the ascent expends of order half a MJ. Getting me to the altitude of low Earth orbit, of order 100 km, would take of order 100 MJ, albeit I'd very much prefer to travel with (at least) an air supply and a pressure vessel to hold it (and me) in, which would increase that a lot; and any fuel I need to take with me for the later part of the lifting needs also to be lifted through the earlier parts, which can add a whole lot more.
The Planck energy, c.c.√(c.h/G), is about 4.9 GJ; this is only slightly more than the 4.184 GJ of chemical energy that one tonne of TNT releases, when detonated. Getting me up to low Earth orbital speed (around 8 km/s) would take a bit over 3 GJ (on top of the 0.1 GJ of lifting, above; and with a matching proviso about air, containment and, especially, fuel).
The energy-equivalent of one gramme of mass is 89.8755179 TJ, roughly the chemical energy released by detonation of 21 kilo-tonnes of TNT.
A magnitude 4 earthquake releases about 1.26 peta-Joules of stress energy in the Earth's crust; this is roughly the chemical energy released by 0.3 mega-tonnes of TNT detonating.