  We present detailed structure and evolution calculations for the first
transiting extrasolar planets discovered by the space-based CoRoT mission.
Comparisons between theoretical and observed radii provide information on the
internal composition of the CoRoT objects. We distinguish three different
categories of planets emerging from these discoveries and from previous
ground-based surveys: (i) planets explained by standard planetary models
including irradiation, (ii) abnormally bloated planets and (iii) massive
objects belonging to the overlapping mass regime between planets and brown
dwarfs. For the second category, we show that tidal heating can explain the
relevant CoRoT objects, providing non-zero eccentricities. We stress that the
usual assumption of a quick circularization of the orbit by tides, as usually
done in transit light curve analysis, is not justified a priori, as suggested
recently by Levrard et al. (2009), and that eccentricity analysis should be
carefully redone for some observations. Finally, special attention is devoted
to CoRoT-3b and to the identification of its very nature: giant planet or brown
dwarf ? The radius determination of this object confirms the theoretical
mass-radius predictions for gaseous bodies in the substellar regime but, given
the present observational uncertainties, does not allow an unambiguous
identification of its very nature. This opens the avenue, however, to an
observational identification of these two distinct astrophysical populations,
brown dwarfs and giant planets, in their overlapping mass range, as done for
the case of the 8 Jupiter-mass object Hat-P-2b. (abridged)
