A nitrogen atom connected to three different groups is enough to make the molecule chiral. In this case, the lone pairs "serve the role" of the fourth group needed to make it chiral. Quote from http://research.cm.utexas.edu/nbauld/CHAPTER%2021.htm:
It is interesting to note that, since the nitrogen atom of amines is tetrahedral, such a nitrogen can be a stereocenter if it has three different R groups attached. By definition, the fourth group is an electron pair, so that all four groups are different.
However, it is observed that when chiral amines are generated, they very rapidly undergo an umbrella-like inversion to generate the corresponding enantiomer, quickly racemizing the amine. Certain amines, for which this inversion is especially difficult, can be prepared and are relatively stable as a single enantiomer.
An atom connected to three different atoms with all the four atoms in the same plane is achiral because the plane of the molecule is a mirror plane hence the molecule and its mirror can overlap.
Please also note there are cases where the chirality is not caused by a simple chirality center, such as helicenes.