Stars evolving along the Asymptotic Giant Branch can become carbon-rich in the final part of their evolution. The detailed description of their spectra has led to the definition of several spectral types, namely: N, SC, J and R types. Up to now, differences among them have been partially established only on the basis of their chemical properties. An accurate determination of the luminosity function (LF) and kinematics together with their chemical properties is extremely important for testing the reliability of theoretical models and establishing on a solid basis the stellar population membership of the different carbon star types. Using Gaia Data Release 2 (Gaia DR2) astrometry, we determine the LF and kinematic properties of a sample of 210 carbon stars with different spectral types in the Solar neighbourhood, including some R-hot stars, with measured parallaxes better than 20%. Their spatial distribution and velocity components are also derived. Furthermore, the use of the infrared Wesenheit function allows us to identify the different spectral types in a Gaia-2MASS diagram. We find that the combined LF of N- and SC-type stars are consistent with a Gaussian distribution peaking at M_bol_~-5.2mag. The resulting LF however shows two tails at lower and higher luminosities more extended than those previously found, indicating that AGB carbon stars with Solar metallicity may reach M_bol_~-6.0mag. This contrasts with the narrower LF derived in Galactic carbon Miras from previous studies.We find that J-type stars are about half a magnitude fainter on average than N- and SC-type stars, while R-hot stars are half a magnitude brighter than previously found although, in any case, fainter by several magnitudes than the rest of carbon types. Part of these differences are due to systematically lower parallaxes measured by Gaia DR2 with respect to Hipparcos ones, in particular for sources with parallax Plx<1mas. The Galactic spatial distribution and velocity components of the N-, SC- and J-type stars are very similar, while about 30% of the R-hot stars in the sample are located at distances larger than ~ 500 pc from the Galactic Plane, and show a significant drift with respect to the local standard of rest. The LF derived for N- and SC-type in the Solar neighbourhood fully agrees with the expected luminosity of stars of 1.5-3M _{sun}_ on the AGB. On a theoretical basis, the existence of an extended low luminosity tail would require a contribution of extrinsic low mass carbon stars, while the high luminosity one would imply that stars with mass up to ~5M _{sun}_ may become carbon stars on the AGB. J-type stars not only differ significantly in their chemical composition with respect to the N- and SC-types but also in their LF, which reinforces the idea that these carbon stars belong to a different type whose origin is still unknown. The derived luminosities of R-hot stars make these stars unlikely to be in the red-clump as previously claimed. On the other hand, the derived spatial distribution and kinematic properties, together with their metallicity, indicate that most of the N-, SC- and J-type stars belong to the thin disc population, while a significant fraction of R-hot stars show characteristics compatible with the thick disc.