If the nucleophile brings the electrons, the electrophile is what receives them. Finding the electrophile in a reaction mixture tells you where the arrow's head goes — and since most organic reactions form exactly one new bond per step, identifying the nucleophile and the electrophile means you have already worked out where that bond appears.
What it is
An electrophile — "electron-loving," though electron-poor is the more useful description — is a species that can accept an electron pair. It qualifies if it has any of three things: a full positive charge, a partial positive charge from an adjacent electronegative atom, or an empty orbital with room for a pair.
Note that a full positive charge is not required, and in most of the course it is absent. The overwhelming majority of electrophilic carbons you will attack are perfectly neutral atoms with a δ+ they acquired from a neighbor.
In acetone, (CH₃)₂C=O, oxygen is much more electronegative than carbon and pulls the C=O bonding electrons toward itself. The carbonyl carbon is left δ+ and the oxygen δ−.
So the carbon is the electrophilic atom — the site a nucleophile attacks — even though the oxygen is the atom that looks more interesting. The oxygen is electron-rich; it gets protonated by acids, not attacked by nucleophiles.
Both atoms have a formal charge of zero. The polarization is entirely an electronegativity effect, which is why formal charge alone is not enough to find electrophiles.
The main families
Polarized carbon. Any carbon bonded to something more electronegative: C–Br, C–Cl, C–OTs, and the carbonyl carbon of every aldehyde, ketone, ester, amide and acid chloride. In CH₃–Br it is the carbon that is electrophilic, not the bromine, even though bromine is the atom that eventually leaves.
Carbocations. A full positive charge combined with a genuinely empty p orbital makes these among the strongest electrophiles in the course. They are the reactive intermediates of SN1, E1, and alkene addition, and their relative stability is the theme of Module 6.
The proton, H⁺. The simplest electrophile there is — no electrons at all, one empty 1s orbital. It grabs the nearest available lone pair or pi bond, which is the whole of acid–base chemistry in Module 3 and the initiating step of a great many mechanisms.
Lewis acids. BF₃ has only six electrons on boron and a genuinely empty p orbital; AlCl₃, FeBr₃ and ZnCl₂ are similar. "Lewis acid" is simply the acid–base vocabulary for an electron-pair acceptor, which is to say, for an electrophile. The same species gets called an electrophile when we are talking about mechanism and a Lewis acid when we are talking about acidity; they are one idea with two names.
Induced-dipole electrophiles. Even a perfectly nonpolar molecule can become electrophilic on approach. Br₂ has no permanent dipole, but as it nears an alkene's electron-rich pi cloud, the repulsion pushes the Br–Br electrons toward the far bromine, inducing a temporary δ+ on the near one. That transient electrophile is what the alkene attacks, and it is why bromine adds to alkenes at all (Module 7).
Making an electrophile stronger
Electrophilicity can be tuned, and synthetic chemistry does this constantly. Protonation is the most common method: protonating a carbonyl oxygen gives a cation whose resonance structures place a full positive charge on the carbon, making it enormously more electrophilic than the neutral ketone. This is why so many carbonyl reactions are run under acid catalysis.
Coordination to a Lewis acid does the same job without a proton, and works where acid would destroy the substrate. Converting to a better leaving group is the third route — turning an alcohol into a tosylate does not change the carbon's charge much but makes the whole system far more reactive toward nucleophiles, for reasons the next section takes up.
Electrophilicity is relative
Nothing is an electrophile or a nucleophile in the abstract; both labels describe a role in a particular reaction. Water has lone pairs and acts as a nucleophile toward a carbocation, but its hydrogens are electrophilic enough to be attacked by a strong base. An alkene is a nucleophile toward Br₂ but becomes electrophilic at the beta carbon once it is conjugated to a carbonyl.
The question is never "is this molecule an electrophile" but "in this pairing, which side is electron-poor." That comparison is what the next section, and the ranking of electron-rich against electron-poor, is for.
What carries forward
The electrophilic carbon of a C–LG bond drives all of Module 6. The electrophilic carbonyl carbon drives Modules 9, 10 and 11. Electrophilic aromatic substitution in Module 13 is named for this idea outright. And the generation of unusually strong electrophiles — acylium ions, bromonium ions, protonated carbonyls — is a recurring theme every time a reaction needs a push.