Most video games you play these days will either have checkpoints and/or savepoints, or lives and continues. Lives and continues are suited for smaller games that have no need for saving if they could be beaten in one sitting. The check/save point method is better suited for large games that can't be beat in under a couple hours, or encourage a lot of customization and personalization, or have many open ended choices.
The lives and continues can be viewed as a mostly outdated format. Game designers would likely have wanted to be able to make a larger and longer game in the early days of video games, and give players the ability to save and come back at a later date and be able to fully experience everything the game would have to offer, but the technology wasn't available. Considering this, the save and load format can be seen as an evolved form of lives and continues. Despite this, some modern games still feature lives.
However, there is one advantage to lives and continues that might have been unintentional, but worked none the less. A game that has no save or load is more difficult to beat. If a game has limited lives and continues, they demand a player advance in skill, allowing only a certain amount of failure before the player has to start the game over, and redo all the easier levels when they want to try again.
A level/world based platformer requiring quick reflexes for example, could get better use out of lives and continues. A stage you die on can be retried a number of times equal to your lives remaining, and when you run out of lives, you might be able to lose a certain amount of progress, in return for one of your continues (your continues being limited or unlimited), and try again.
An opened ended Sandbox or RPG has no need for lives or continues, the game genre typically being so massive that being able to save your progress is a must to be able to beat the game, and the blow of dying or failure in the game is softened by being able to infinitely return to earlier state by loading your save. Other games that work better with this system are games that are just too long or too hard, that beating without taking a break is unheard for a typical gamer.
However, nearly all games these days have a save feature, a function many early games did not. Older games have even frequently been redesigned and updated, having a save feature embedded into them. This is where the problems begin. The two just don't mix. If a game that has lives and continues, and a save feature, lives become meaningless. If a player runs out, he can simply restart the game and load his save, causing the outdated feature to come into questioning. This problem is notorious in recent Mario titles. This has caused lives to be viewed as mostly meaningless, so I will close with an overview of how lives and continues works in Super Mario Bros. 3 for the NES, and in my opinion, it's one of the best games to use the lives and continues system perfectly.
Every time you clear a level on the overworld map, that's a "checkpoint". If you die, you lose a life, and you're sent back to that location, having the option sometimes to head down a different path, or just to try again on the level you just died in. While going through levels, you're receiving more lives, and your inventory is being equipt with items you can later use. If you beat one of the 8 worlds, that's both a "checkpoint" for your lives, and for your continues. If you do run out of lives, you can continue from the last world "checkpoint", and you get a few lives to work with, allowing you to continue an unlimited number of times, at the cost of all your progress up until that point. The game is basically telling you "alright, go back and try these easier parts again, visit the Toad Houses and Spade Panels again, stockpile some lives, hopefully you'll do better this time".
I hope this personal take on things was helpful to you in some way. Thank you very much for reading!