Shinobi have many duties in Shogun: Total War. Ranging from spying to policing.

1. Spies - When shinobi are put into enemy provinces and left there for one turn they are spying for you. After being there for a turn the shinobi give you pretty much every bit of info you need on a province. They allow you to see how many and what types of troops your enemy has in the province. They also show you the facilities the enemy has there.
2. Counterspies - When shinobi are left in a province you have control of they are counter-spying. By counter spying a shinobi can detect and stop enemy ninja.
3. Policing - When shinobi are kept at home they also raise the population loyalty in that province. They help keep any rebellion talk to a minimum.
4. Inciting - When a shinobi is placed in an enemy territory he immediately has a an effect on the enemy population loyalty. This can lead to revolts.

That is the basic functionality of the shinobi. However there is always more to a unit then their mere basic functionality. There are many applications of shinobi that may not stand out.

First and foremost shinobi are great spies. They gather all the information that an emissary can collect. However, unlike the emissary they are not as easily detectable. This makes catching them and stopping them much harder from your enemy's standpoint. This also makes them of much greater value to you.

Another use for shinobi is as counterspies. By having shinobi in a province, enemy ninjas are less likely to be able to assassinate anybody. A good tip here is to keep one or two shinobi near your important people. Your Daimyo should have a small army of his own shinobi as he will be the main target of high-level ninja assassinations. Your heirs are the next, most tasty targets for ninja so provide them with a shinobi or two. High-ranking Generals are another prime target for ninja so don't forget to give them a shinobi or two for their own protection. If an enemy can't beat your general on the battlefield he will try to assassinate him. If you lose a General before a battle your troops will suffer a lowered morale thus making them easier to rout. I don't think I need to explain the importance of keeping your heirs and Daimyo safe!

An interesting feature of the shinobi is his ability to change population loyalty wherever he goes. A very neat tactic used to exploit this fact is the "swarming shinobi". Send an "army" of shinobi and I mean 10 or so into an enemy province. This will cause the province population loyalty to go way down. It may even go down enough to cause a revolt in that territory. Now your enemy must spend some of his time and army squashing the revolt. You can either attack at the same time that the revolt is going on, thus the enemy will have 2 enemies to fight. Or you can wait until it's over with and then attack. Either way the enemy force is a little weaker now, so this tactic can provide a little boost to your army in that respect. It also has another side effect. If you keep the "army" of shinobi in a province you are about to take over it will make keeping the loyalty up when you do take it over much easier. If you take over a province who had a loyalty of 300%+ you are going to have to keep a large garrison there to keep that population happy. If you attack and take a province with a very low population loyalty, it will be much easier to get their loyalty up after your arrival requiring a much smaller garrison.

A variation to the tactic above is to use the shinobi more defensively. This works well when using the satellite defensive set up. If you don't know what this is I'll quickly explain. You have a center main province with a huge army and the surrounding provinces have one unit of Ashigaru in them or some other small unit. When the computer attacks he will only bring a small force to counter your small force. Here's where the shinobi army comes in. Withdraw your small unit from the attacked province and send in your entire shinobi army. The population will immediately revolt against the enemy. Now you can either move in a small sized army to help the uprising out or you can just wait and see how the peasant do against the enemy then attack. This is a way of getting free troops to fight for you while maintaining one small army at the center of your surrounding provinces instead of having a bunch of armies.

Another way I use shinobi is when I am conquering a certain area with a small force. Let me just take for instance the rebel island below the Mori land. There are 4 territories with a troop production facility in Tosa. The main goal when attacking from the eastern side of the island is to take Tosa out so they cannot produce any more troops. So I had very little army to spend on this task so I bought 3 warrior Monk units and two shinobi. Once I took Sanuki I move to Iyo. Now I don't have a large enough force to keep a garrison behind me so I had to move on. I left a shinobi in Sanuki as I move my three-unit force into Iyo. Well much to my surprise the shinobi in Sanuki reported to me that the peasants planned to revolt and they planned to do it next summer. This information was invaluable as it gave me a timetable until my path off the island behind me would be closed off. I knew I had to take out the island in less than four turns or else I would have to bring in troops to rescue my three-unit army that was on the island. This information is invaluable when conquering territory with a small army.

Shinobi also provide you with a cheap garrison. Even in your own territories it's good to keep a garrison to keep the loyalty at a decent level. You can use troops for this like Ashigaru or even Samurai. But why tie up a season of spear troop production just to maintain a garrison when you can use a shinobi. Shinobi are just as cheap and do just as good a job in already conquered territories.

Border Towers/Border Forts vs. Shinobi
Border Forts act as spies in the territories surrounding that province. Border Towers act as a both a spy and a counterspy. So which is better Forts/Towers or Shinobi?

It depends on the situation. Shinobi of course have the advantage of being mobile. They can move beyond a territory away and give information anywhere they are. The only problem is that shinobi can die; enemy security forces can catch them. But to combat this, shinobi are cheap to make so replacing them isn't that bad of a deal.

Use Forts/Towers when you have a territory that has more than two enemy provinces on its border and you do not intend to expand into that area any time soon. The reason for this is that if you build Forts/Towers and you plan to move into the territories you will have to build them again once you do. This is not very economically sound. Also, if there is only one enemy territory on your border just build one shinobi and put him there. Even if there are two enemy territories just build 2 shinobi. When you have a province like Shinano that has 10 borders you might want to put up a Fort/Tower. Since its better than keeping track of and building 10 shinobi.

Use Shinobi on the fronts you are expanding through. This way your spy/counterspy can move with you wherever you go. As mentioned above they give information on peasant revolts, decrease the population loyalty of the enemy you are about to attack and protect those high level Generals that you are using to fight these battles.

Addendum:
This little bit of information was given to me by Kingsley Gardner, kgardner@tycoint.com
Border Towers will only catch one shinobi a turn. This is important information when one is planning to infiltrate a neighboring enemy territory, which has a Border Tower. To get around a Border Tower simply send in more than one shinobi a turn. The first shinobi will be caught but the rest will move on. Thanks.