Since you don't provide an age, there are a few different guidelines based on age.
For children under about eight or nine months, stories primarily serve the purpose of hearing the parent's voice, as a soothing influence and helping to develop language. Therefore, any story will do. Books primarily should be high contrast, black and white in particular or very bright colors with few variations (ie, an entirely green turtle, not one with various lines of different colors that is more realistic).
Between nine months and two years or so, they often enjoy books with pictures (in particular, photographs) that they can identify. Books with lots of pictures of trains or trucks if they're a transportation kid, animals if that's their thing, etc.; ABC, color, or number books; or "Baby's First Word" books are all good here. Stories in terms of plotted stories won't be very effective, because they won't understand the concept of plot or know why they should listen, and they don't necessarily want to go in order.
After about two and a half years, varying based on the child, you can start having plot-intensive stories. Fairy tales, Curious George or Clifford or other characters, Dr Seuss, etc. are all fair game. They'll like to listen to the story, and begin to be able to discuss some of the elements of the story. Some children will even begin to role-play around 3 based on the stories; my three year old likes to make up stories of his own, where he's a main character alongside Curious George or similar.
Once they hit the age that 'plot' stories make sense, and you manage to find out what they enjoy (by trial and error, most likely), I like the idea of taking formulaic stories that they enjoy book versions of, and make up your own versions. This doesn't require any preparation other than knowing what the formula is from having read quite a few. My son likes Curious George; for that, the formula is:
- "This is George. He was a good little monkey, and always very curious."
- George is [somewhere] - establish the location
- George is doing [something] or going [somewhere else]
- George is curious about [something] and does [something]
- This leads to trouble
- George is sorry he got in trouble
- George manages to get out of trouble
- The trouble turns out to have been for the better because of [some reason]
- George is happy, the end
You can make that up that 'on the go' pretty easily, even if you're not creative (I am absolutely not!). Several of the other book series work similarly, as well as Aesop's Fables and some of Grimm's fairy tales: there is a fairly simple formula that anyone can follow, just filling in the details around the formula. The nice thing is that kids don't care too much about the actual quality of writing - so just make it customized to the kid, and you're all square.
For example, one story I'd tell my son (let's call him R) goes something like this:
This is George and R. They were good little guys, and always very
curious. One day, George and R were going to the store. The Man with
the Yellow Hat came by to pick R up, and he was very happy to get in
the blue car and go. They drove up to the store, and R and George got
out. After they went in the store, they got a shopping cart, and R
got in. George pushed R around in the cart for a while, then they
The Man with the Yellow Hat needed Sugar, Baking Powder, Chocolate
Chips, and flour, to bake the cookies they wanted to bake. While they
went past the cookie aisle, George saw something he liked, and ran
off. R followed him, and said "No George, don't run away!" George
stopped, and saw a little boy crying in the aisle. R couldn't find
the Man with the Yellow Hat, so he found a worker at the store, and
told him the boy was missing his mommy.
The worker paged the boy's mommy over the intercom, and his mommy came
by a little later, and everyone was happy. The Man with the Yellow
Hat also came, and told George that he shouldn't run off, but that he
was happy George and R knew what to do to help the little boy. They
went to the checkout and bought the ingredients, then went home and
baked cookies together. R got to eat the first one out of the oven,
and it was very yummy
I wrote that in about 3 minutes, and largely copied off of a similar story I read frequently (George goes to the Baseball Game, or something titled like that). It has elements of my son in it - he likes to run down aisles, like every kid - and elements I like to teach him (asking a store worker if he's lost). It also has a lot of 'hooks' for him to interject details - for example, he can provide the details of the shopping list, and what they're making at home, or the store's name, or what aisle George ran down. It's also a terrible story from an adult perspective, but it's loads of fun for a three year old.
What's nice about doing it this way is not only that you have a story wherever you want, and you involve the child; but also that it encourages the child to make up his/her own stories (both by involving him/her in the process, and by modeling the process). I found that more and more of the story would be made up by my son in this process as we went along - starting by just adding a few hooks for him to add, but as we went on he'd insist on more and more details, and as long as I didn't resist too much it worked well.
Adding a bit for older kids, there are formulas you can use even for quite complex stories. If your kids like Tolkien, there is a common Tolkien-ish formula. Here's my approximation of it:
- Unassuming hero learns he is the key to saving the world from something
- Hero finds a group of friends, each of which adds a different element to help him along
- Hero and friends must take a long journey to [find|destroy] something which will [prevent|enable] the bad guy from [destroying|conquering] the world
- Bad guy sends various enemies at the hero's party
- Relatively early on the mentor either dies or is removed from action, forcing the hero to grow up
- Hero makes mistakes, which cause a visible change in the hero and showing he is growing up/becoming less naive
- Major battle occurs, forcing the hero to go it alone (or almost-alone) for the final part
- Hero accomplishes the goal, but not seeing his life change
You could watch Star Wars, read (or watch!) Lord of the Rings, or read countless fantasy books and see this same formula used over and over again. Some stick more closely to it - see Terry Brooks or David Eddings, both of whom unabashedly stick to it fairly tightly - perhaps the closest of which is Star Wars itself. I am not going to replicate this here due to space and time constraints (and I'm pretty terrible as an author), but it's not all that hard to replicate - and in fact, many D&D games that are more on the storytelling side end up replicating this (often intentionally). By 8 or 10 you could easily do something like this with your children, and involve them (perhaps with some roleplaying aspects, perhaps not) and have a lot of fun.
There are also other formulas; famously, most romance novels published by Silhouette or Harlequin are explicitly formulaic (they are expected to follow certain formulas). See Wikipedia for some details and examples.