The terms advection and convection have different meanings depending on in which context they are used.
In fluid dynamics, advection is a type of convection, with the other type of convection being diffusion. Both advection and diffusion act to move around various intensive properties of a fluid.
In advection, properties are transported by bulk motion of the fluid. In other words, as soon as the velocity field of the fluid is non-zero, you have convection of the various intensive properties of the fluid, because they are moving along the velocity field.
For example, if you pour some dye into a river so that a colored blob is formed in the river, that blob is going to follow along with the river; the movement of the blob along the river is a form of advection.
In diffusion, things are spread out from points of high concentration or high property values to points of low concentration or low property values, as a result of particle random walk, which causes nearby regions of the fluid to blend together and has the effect of "blurring" the the various properties of the fluid. Mathematically, this means that high frequency components of the properties tend to decay over time, with components with higher frequencies decaying faster than components with lower frequencies.
When modelling turbulent flow, a common practical strategy is to ignore the small-scale vortices, or eddies, and to instead model their large-scale effects on various properties as a type of effective diffusion.
For example, thermal conduction is a form of diffusion, where the diffused property is the temperature. The effect of viscosity is another example of diffusion, where the diffused property is the velocity. For turbulent flow, the mixing of the velocity field caused by the eddies is sometimes modeled as an effective diffusion called the "eddy viscosity."
And to take an example that uses dye to explain diffusion, if you add a drop of dye carefully in a glass of water, you may form a very small blob of dye on the surface of the water, with a clear boundary towards the water. As time progresses, though, even if the water seems to perfectly still, that dye is going bleed out into the water, and the crisp boundary between the dye and the water is going to become blurry end eventually disapear. This is because even if the water is still on the macroscopic level, on the microscopic level the molecules move around and mix with each other, and dye molecules will become mixed with nearby water molecules. This movement of the particles on the microscopic level causes a blurring effect on the macroscopic level and is a form of diffusion.
In meterology, as pointed out in Kobus' answer, advection means horizontal winds, and convection means vertical winds. Convection is typically caused by a steep lapse rate, which gives rise to a Rayleigh–Taylor instability in the atmosphere.