Aioli is a mayonnaise with garlic, with or without vinegar or lemon juice.
It's fairly easy to make, and I expect that everybody has their own recipe, but here's one that I quite liked, served with steaming new potatoes and onions, to go with salmon (gravad or smoked), your choice of bbq meat, or whatever else you fancy:
Take 1 egg (or 1 egg yolk)
add 1 tablespoon of vinegar (I like white wine vinegar)
add 1-4 garlic cloves, chopped up
(don't zap it in a blender: the garlic is much too strong even for some garlic lovers, if you do that)
then, first drop by drop, then in a thin stream, then in a less thin stream, add 1 dl of a cheap cooking oil while beating the mixture all the time.
Add salt to taste.
You can make more by adding egg (or yolk), vinegar, garlic and oil. Adding just garlic and oil works, too.
Or another recipe:
Take 1 egg yolk
add a cheap cooking oil as above - you can make a liter of aioli from one egg yolk if you so wish
add garlic to taste, chopped-up or pressed
add spices: curry, salt, pepper, sugar, soy sauce, mustard, fresh chopped herbs ...
It's best to use the cheapest of all oils for this, tasteless, bland, industrial-strength: the taste of finer cold-pressed oils comes through strongly enough to ruin the taste of your aioli.