A measurement of the underlying event (UE) activity in proton-proton collisions is performed using events with charged-particle jets produced in the central pseudorapidity region (|η jet| < 2) and with transverse momentum 1 ≤ p T jet < 100 GeV. The analysis uses a data sample collected at a centre-of-mass energy of 2.76 TeV with the CMS experiment at the LHC. The UE activity is measured as a function of p T jet in terms of the average multiplicity and scalar sum of transverse momenta (p T) of charged particles, with |η| < 2 and p T > 0.5 GeV, in the azimuthal region transverse to the highest p T jet direction. By further dividing the transverse region into two regions of smaller and larger activity, various components of the UE activity are separated. The measurements are compared to previous results at 0.9 and 7 TeV, and to predictions of several Monte Carlo event generators, providing constraints on the modelling of the UE dynamics.