To cultivate is to turn, move, mix, stir or tidy the soil. Cultivation is the preparation of soil for growing plants. Weeding is an important part of cultivation - weeds grow fast and reproduce fast so if not exterminated by the diligent gardener they will deprive your plants of both nutrients and water.
The type of cultivation required depends on the type of plants which will be grown. New ground should first be dug with a fork, so that weeds can be removed in their entirity. A spade tends to break up, and propagate, weeds. Treading or rolling the soil breaks down lumps and stops the soil being 'puffy'. Surface lumps of soil are further broken down with a rake. Seeds require a fine tilth, free of weeds and stones.
A hoe is not used as part of the initial cultivation job: it is a specialised tool for removing weeds between growing plants. Hoeing works best is hot sunny weather, because it exposes roots to descication. If used in wet weather the weeds are likely to survive by putting down new roots.