There are traditionally two groups of flowering plants - monocotyledons and dicotyledons. It has turned out that dicots are not a "real" group and are just every flowering plant which isn't a monocot, but the distinction is still useful. The true dicots are, unsurprisingly, referred to as "eudicotyledons". This raises philosophical points about clades and taxonomy.
Monocots are never trees but include some tree-like plants, often ones native to the tropics such as palm trees, bamboo and banana trees. This is due to the position of the vessels in their stems, which makes it impossible for them to grow via secondary thickening. They have parallel, non-branching veins in their leaves, their flowers have three or six petals, they have fibrous roots and their seeds have single embryonic leaves and therefore form one part. Dicots can be trees, although not all trees are flowering plants, they have taproots, their flowers have four, five or ten petals (not always, e.g. the cabbage family) and their seeds generally have two halves. However, there are many exceptions to each of these. For instance, the dicot plantains have parallel venation (not to be confused with bananas), waterlilies have lobed cotyledons which may or may not be two fused cotyledons and there are also basal "paleoherbs", which are flowering plants which diverged before the distinction evolved.
One distinction i've not mentioned here is in the pollen, which has three furrows in dicots and one in monocots.
I feel weird about monocots because they sort of feel more alien than dicots. If someone asked me to name a typical flowering plant, i would probably name a dicot and the monocots all seem a bit samey to me. But that's just me.
Also, a whole load of weird stuff happened in plant classification in the past few decades which is very confusing.