The meta description is not visible in any part of your site, except if you check the actual HTML code view. It should be used excusively to supply robots (from search engines like Google) with a short summary of the page. If set up correctly it iwill indeed be used by these search engines as the snippet found in the Search Engine result Pages (SERP's) to describe a search result for a webpage. You can also see it as the sales pitch to your website, as it can trigger people to actually click on your result in Google:

Actually, the meta description is NOT used as a ranking factor, which sometimes causes people to think it is not important from an SEO perspective. However, their impact on Click Through Rates can be massive. The best way to describe its importance is that you can see it as a little advertisement for your website, for free... Similar to the page title, attractive meta descriptions are very important and make people click on your site.
The only thing Google hates is duplicate meta descriptions that are re-used in many pages. They even stated on their webmaster blog that you are better off having empty meta descriptions then duplicate ones. If you do not enter the meta description for your pages, Google will just try and take a snippet from the text of your page. Usually, it will do a pretty good job in doing so, except that it will usually look a bit messy, with sentences broken off, and the use of '....'.
However, if you enter the meta-description yourself, you have full control of this text, as long as it is relevant for your search results. If a user searches for the word "SEO", but the meta description you filled in does not contain this word, Google will probably ignore your description and use it's own. Using a correct meta-description has the following advantages:
- If you make sure you limit the description to 160 characters, it will not be broken off, and won't look messy.
- If carefully composed, users browsing through the search results may be more likely to click you site, increasing the Click Through Rate (CTR), which in itself can be positive for your SEO ranking.
- Google does NOT use meta descriptions to rank you higher. However, if set-up correctly, Google might register the higher CTR and reward you for this.
- Google hates duplicate meta descriptions.
- If the keywords that people search for in Google are also used in your meta description, it will be made bold in the search result pages.
There has been much debate about optimal lengths for the meta description. Untill November 2017 Google used a pretty fixed limit of around 160 characters (actually, Google uses pixelcounts, but that is hard to work with), but then they expanded it to around 320. For half a year... This led many people te rewrite their descriptions, only to find that they had to go back 6 monts later.... I personally think this was a very awkward move by Google, but nevermind. They currently say there aren't any formal requirements and indeed meta descriptions vary depending on the keywords serached for, but in 90% of cases, the old limit is back. So, the best advise to give is this one:
Make sure to sumarize the most important information in the first 160 characters. Feel free to add more for the rare occasion it may be shown, but don't spend too much effort on this. Finally, Google has become very picky with your meta descriptions. You really need to get it right for Google to re-use it.
Joomla setup: Setting a global meta-description or not
When building a Joomla site, the first thing you are often advised to do is to add a global meta-description, using the Global Configuration settings. This ensures that every page will have a meta-description. However, it will be the same for all pages of your site, which is why I personally leave it empty (I add it for individual pages later) and I strongly advise you to do the same. If you want to use it though, you can enable it from:
System > Global Configuration > Site tab, under Meta-Data
In the Site Meta Description field, fill in a useful description, using your main keywords:

Meta-description for every page
You should make sure that every page has a unique description. Unique is really the keyword here! Then every page can build its own ranking for its own specific keyword or keyword-combintion. If you have a really large site, this may be a lot of work. In cases like that, you may have to prioritize: start with the homepage and the main menu-items and work down as far as you have time.
You can set your descriptions either per article, category or menu-level. If you have a simple site with only a few articles, where every article is linked to a menu-item, it does not really matter which one you choose. On larger sites, or sites using blog-views, not every page is connected to a menu-item. Probably the best way to go is to set it in every article, for category views per category, and for the remaining ones on menu-level.
The most used method is the Article Manager. In ebery article, you will find the Meta Description field under the Publishing tab. Similarly, in the Menu Manager you will find it under the Metadata Options:

Note that if both are entered, the menu-item description will be used, overriding the one you set on article-level.
Joomla extensions
There are many extensions to assit you with your metadescriptions. In the article about Page Titles (at the bottom) I list a few, and these are the same as the ones to be used for meta descriptions.
Pro-tip: It is always wise to perform an SEO audit that also scans whether your meta descriptions are indeed set-up correctly. I can highly recommend Ryte.com for this. It's free for one website (read my review of this service). For more sites, I personally also use Website Auditor by SEO Powersuite (desktop, free if you don not need to save projects).