Disclaimer:
These problems are meant to be examples of the common ways chimneys can leak from my past experiences. Any of these examples or none of these examples may apply to your chimney. Always contact a licensed chimney expert for your particular chimney leaking problems.
One of the most pesky ways a chimney will leak is through the mortar joints or brick/stone/block. When the chimney exterior has bad or damaged masonry, water can enter in through the holes or cracks in the joints. When sufficient water has entered in, gravity takes over and pulls the water down through the chimney. Most chimney flashings are cut into the brick work about an inch into the masonry, but if water is penetrating deeper than the flashing, the water will bypass the flashing and can enter the house. Chimneys with poorly or incorrect flashings are even worse, stopping virtually none of the water from entering the house.
As more water enters the chimney, the more deterioration of the brick and mortar occurs, especially in northern environments where the winters can be cold and harsh. This can lead to an exponential deterioration process of the chimney costing you more money in the future to fix.
If the structure of the chimney is solid and the mortar joints need some repair, sometimes your licensed chimney expert will recommend repointing the chimney. This consists of removing the defective mortar joints to a certain depth either mechanically or manually. Then once the mortar has been removed, the open joints are then refilled. When the mortar is starting to get stiff, often the joints will then be "stuck" or "tooled" to give it a finished look. Tooling the mortar joints properly will compress the mortar into place.
If the structure of the chimney is not good, it is often recommended to remove the chimney to a solid point in the structure and then rebuild it with new brick and mortar.