One of the most important element of Joomla are the components (extension). Components can be considered as mini-applications inside the Joomla ecosystem, that in most cases are designed to allow the creation, management and display of content for the site. There are many core components and a lot more 3rd party.
The most used component is the com_content (articles). It is the one that let us create categories and articles.
Another important element of Joomla, is the Menu Items. Menu items are responsible for displaying content and the overall structure of a Joomla site.
Each component provide a set of menu items with predefined settings and layout options. We create menu items to link certain content from a component.
An example of a menu item of the com_content component is the Single Article menu item. With this menu item, we can create a menu item in our menu, to create a page that will display a specific article.
With the com_content component we can create other types of pages as well, like a blog layout page. For a such page, we can use the Category Blog layout menu item to display many articles, one below the other of a certain category.
Now as said, each menu item has settings and layout options. The category blog menu item for example gives us options to select how many articles we would like to display in each page before the pagination starts.
We have options for leading articles, intro articles, columns (usually for the intro articles, but that depends also on the template) and link (only) articles.
See the screenshot...
You can experiment with these settings for the menu items of your categories and see how they change on the front-end.
Another thing to note is that these options can be set as global, to be used as default across the com_content. To access the global settings, just click on the articles and when on the articles list, click the Options button (left on the toolbar).
Regarding the case that some articles are shown with only the first 1-2 paragraphs, and clicking on the title takes you to the full article, this in Joomla can be achieved by using the Read More feature inside an article. There should be a special button, below the editor textarea which will insert a special
<hr> tag. This will instruct Joomla to split the whole article into 2 parts. The intro text and the full text. Only an article that contain this special "Read More" will be shown half in a category blog layout and full or the remaining part in the article page (depending on the chosen settings for intro-text).
Well, there are a lot more that can be said regarding options, features, possibilities etc... but I won't expand to cover each and every case. I hope the above are a good introduction.
I just want to note again that in many cases, the templates and possible overrides can have their own touch on how the content is displayed in the front-end.
For now I would suggest you to go and explore your site and its structure, review your global settings, menu items and articles, and experiment with the settings. If you need further help or have other questions you can come back here and ask a new question.