The paper presents new concepts and results for the eddy structure of turbulent convection in a horizontal fluid layer of depth $h$ which lies above a solid base with thickness $h_{b}$. The fluid parameters are the kinematic viscosity $\nu $, the thermal diffusivity $\kappa $, which is taken to be comparable with $\nu$, the density $\rho $, the specific heat $c_{p}$ and the expansion parameter $\beta$. The thermal diffusivity of the solid is $\kappa_{b}$. The results are an extension of the more commonly studied cases, where a constant heat flux or constant temperature is applied at the interface between the fluid and the base. The buoyancy forces induce eddy motions with a typical velocity $w_{\ast} \sim (g \beta F_{\theta} h)^{1/3}$ where $\rho c_{p}F_{\theta}$ is the average heat flux and $F_{\theta}$ the covariance of the fluctuations of the temperature and of the vertical velocity. At moderate Reynolds numbers ($Re=w_{\ast}h/\nu $), say less than about $10^{3}$, an order-of-magnitude analysis shows that for the case of high diffusivity of the base (i.e. $\kappa_{b} \gg \kappa$) elongated ‘plumes’ form at the surface and extend to the top of the fluid layer. When the base diffusivity is low (i.e. $\kappa_{b} \leq \kappa$) the surface cools below the developing ‘plume’ and either the plume breaks up into elongated puffs or, if $\kappa_{b} \ll \kappa$, horizontal pressure gradients form so that only small-scale puffs can form near the surface. At very high Reynolds numbers, approximately greater than $10^{4}$, the surface boundary layer below each puff/plume is highly turbulent with a local logarithmic velocity and temperature profile. An approximate analysis indicates for this case that there is insufficient buoyancy flux from the base, irrespective of its diffusivity, to maintain plumes, because of the high turbulent heat transfer. So puffs dominate high-Reynolds-number thermal convection as numerical simulations and field experiments demonstrate. However, when the surface heat flux is uniform, for example as a result of radiant heat transfer or by forcing with a constant heat flux below a very thin conducting base, plumes are the dominant form of eddy motion, as is commonly observed. In the numerical solutions presented here, where $Re \sim 3 \times 10^{2}$ and the slab thickness $h_{b} = h$, it is shown that the spatial scales of eddy structures in the fluid layer close to the surface become significantly smaller as $\kappa_{b}/\kappa$ is reduced from 100 to 0.1. At the same time in the core of the convective layer the change in the autocorrelation and spatial correlation function indicates that there is a transition from long-duration plumes into shorter-duration and smaller-length-scale elongated puffs. The simulations show that the largest temperature fluctuations near the surface occur when a constant heat flux is applied at the bottom of the fluid layer. The smallest temperature fluctuations are associated with the constant-temperature boundary condition. The finite base diffusivity cases lie in between these limits, with the largest fluctuations occurring when the thermal diffusivity of the base is small. The hypothesis introduced above has been tested qualitatively in a laboratory set-up when the effective diffusivity of the base was varied. The flow structure was observed as it changed from being characterized by nearly steady plumes, into unsteady plumes and finally into puffs when the thickness of the conducting base was first increased and then the diffusivity was decreased.