Some of the other answers cover the winds on the open ocean. Those winds are more or less constant; the same winds go in the same direction the same time of the year. Prominent (historically) among winds are the Atlantic North Equatorial current and corresponding winds, which runs straight across the Atlantic from the Canaries to the Caribbean and took the initial explorers and conquistadors to the New World, and the season spring-fall monsoon winds of the Western Indian Ocean which moved ship commerce between India and Arabia and East Africa.
However, the biggest cause of delays were port conditions. There are three factors to consider.
There were generally two types of rigging, square and lateen. Lateen rigging allows a ship to tack 'into' the wind, allowing it to make slow progress directly against an oncoming wind. It also improves maneuverability. However, it provides poorer performance 'running', that is going with the wind 80 degrees or more abaft the beam. A running lateen sail's yard will be extended outwards over one side of the ship, to orient the sail perpendicular to the wind. However, if the wind shifts direction by a few degrees, the forces can attempt to switch the side that the yard wants to project on. A rapid shift of the yard will cause the ship to 'back' removing forward momentum; the swinging yard can kill passengers, destroy rigging, and even flip the boat.
A square rig is much safer, and better at catching the wind too, when running. All the big ocean-going tall-ships of the Age of Exploration depend on the square-rigged sail to cross oceans; it provides more thrust per sail area, less rigging to handle, and is stable in conditions of steady following winds.
Lateen rigging is preferred in more geographically compacted seas with more variable winds, such as the Mediterranean. In the Indian ocean, the Arab dhows with their lateen sails were quickly driven out when the Europeans showed up with their larger square-rigged sails.
The big advantage of a lateen rigged ship is that you can get into an out of a port under many conditions. Since a port is designed to be protected from ocean swells, it is almost always in some sheltered location with limited access to the open ocean. If there is only one direction you can go to get to the open ocean, then it is entirely possible that there is just not enough wind to get there.
Lateen rigged ships are much more likely to be able to move in the direction they need to go to get out of port. Lateen rigged ships can often start from anchorage in a 180 degree arc along the direction of the wind. A square rigged ship's ability to start is more like 90 degrees. Further complicating matters, is that it is important which way you are facing when you try to start sailing. If a square rigged ship is facing into the wind because of the direction of the current where it is anchored, then it cannot raise its sails at all, or it will start drifting immediately. I won't get into too much detail, but a lateen rigged ship has a lot more options.
This brings us to the last point, tides. Under certain tidal conditions, you would not want to get underway. Certain tidal estuaries can have very high currents at different points in the tidal cycle. Examples that I have seen personally include Lisbon (5 knot currents in the estuary) and Lagos, Nigeria (where I had to order an ahead bell while mooring due to extremely high currents in the lagoon). Despite that fact that I sailed a ship under power, with tug assistance, these operations were both dangerous and would have been better avoided.
I don't see how a sailing ship could attempt to get underway in such conditions. Thus, in certain ports, only at high or low tide, when the current is slack, can a ship expect to get underway safely. Other ports, especially those with man-made barriers such as the ancient ports at Carthage or Alexandria, have shallow bars that develop at their entrances. These bars mean that a ship can only expect to get underway at high tide. Thus, depending on the situation, a ship may only be able to leave port at high tide.
The real problem is the combination of tide and winds. If you can only leave port at high tide, then you would get your crew aboard every day at high tide and see if the wind was right. If it wasn't, you weren't leaving that day (its not like you'd get underway at night). Send the crew ashore for some drinking and wenching, try again tomorrow. This could go on for weeks.
Coastal sailing is not done at night in the ancient world. Perhaps if the moon is full, but generally there is too much risk in running aground. If you want to keep going at night, you have to get out into open sea. That means coast-huggers use lots of temporary anchorages (or haul-outs where they beach the ship, if it is a galley, see The Odyssey) while hugging the coast from point A to point B. A wise captain will know the winds and tides and predict good places to hang out overnight so he can start right away in the morning. A foolish, inexperienced, or unlucky captain will not.
Roman records in particular are filled with travelogues of the varying calamities that can cause delays. Most prominent among them are delays leaving port, and delays caused by being trapped in a temporary anchorage.