Lowest Cost used to be called "Automatic Bid". Lowest Cost with Bid Cap is identical to what was previously known as "max bid". Lowest Cost with Cost Cap used to be called "average bid".
See also: thorough explanations of all the Bid Strategies offered by Facebook
Recommended: Lowest Cost
Facebook’s Lowest Cost (without a bid cap) is the recommended bidding option when you simply want to spend your whole budget optimally. It means you don't have to spend time managing your bids, but you can simply steer your spending with budgets. It makes sense in following situations:
1) You have a certain budget to spend, and you want to spend the whole budget, optimally
2) If you just want to get maximal amount of the targeted actions for your budget and let Facebook take care of bidding
NOTE: Lowest Cost cannot be used without a bid cap when using App Installs billing event.
Bid limited delivery: Lowest Cost with Bid Cap
Lowest Cost with Bid Cap comes handy when your goal is not to spend your whole budget, but rather control the cost of each conversion. Using Bid Caps makes sense when:
1) You know how much exactly you are willing to pay per a Facebook-reported action, such as a conversion event
2) It's not so important to spend your budget fully, but rather spend as much as possible while under a given KPI goal
A Bid Cap informs Facebook how much is the worth of one action defined in Optimization goal to you = how much you are willing to pay per action at most.
Remember that the Bid Cap is the maximum value per each individual conversion. Facebook will also deliver to people who will likely convert at a lower price than that. As a result, the average price per conversion will be lower than your Bid Cap.
As the Bid Cap is linked to the Optimization Goal and conversion event of your ad set, so don't forget to adjust the Bid Cap if you change the Optimization Goal or conversion event.
Read more: How much should you bid?