FPS Shooter (prefer Snipers)

I don't have any suggestions here that haven't already been made.

Puzzle

• Oddworld: Abe's Exoddus, Oddworld: Abe's Oddyssey -- If you get any puzzle games, both of these would be my personal recommendation.

Fantastic, fantastic games. It's hard to give a basic synopsis for them but they're from the PS1 era, so you should probably be okay. They are brutally difficult, however.

• Gyromancer

Simple and easy to get into. A sort of Bejeweled/RPG hybrid with tons of content. Cheesy Japanese RPG story bullshit abounds as it's a relatively low-budget Square-Enix game, so be ready for that if you get into the game.

Open World/Exploration -- This genre is really, really hard to recommend as any of these games are likely to be taxing on a computer.

• Mount & Blade: Warband

A medieval open world map with first/third person combat. It isn't much of an exploration game but it has absurd amounts of replay value and is relatively low impact on systems. I recommend Warband over the original game, however. It's just better in general. There isn't an overarching narrative however. It is a sandbox with a framework, nothing more. That said, some VERY good mods are available on that front.

• Bully

GTA-style game but with a very unique take on play and story compared to them. It's from the PS2/Xbox era, so it may or may not be too much for your laptop.

Horror or Survival

• Legend of Grimrock

Old-school 4 character first person dungeon crawler. There is also a sequel but it has higher PC requirements, so I can't guarantee success on running it. These games are extremely high quality and well-made. However, they ARE niche. There are multiple playthroughs of the games available on YouTube, so maybe watch a video or two to decide if you want to get into them.

• Don't Starve

Isometric survival/horror game. Interesting art style with a HUGE amount of content. If you ever get better Internet, there is a co-op version in development.

Simulation or Strategy

• Warhammer 40k: Dawn of War 2

Squad-based RTS. The numerous expansions add up to a rather painful price tag for such old games, however. The Ork campaign has some of the best dialogue ever crafted for a game.

• The Banner Saga -- If you feel your laptop can handle it, this is probably one of the best games on my list.

More modern example of a tactical RPG game, might be too much for your system. However, if you think your laptop is up for it, it's probably in the top three of best indie games ever made.

and since its a laptop please recommend older games

That really doesn't give a good guideline for what it's capable of handling. My laptop is only 6 months old but I wouldn't play much more than a heavily-limited Civ 5 on it if I ran games on it at all.

That said, a laptop not really built for gaming has an obvious solution for games, Rogue-likes and isometric hack & slash games.

Rogue-likes

1. Binding of Isaac: Rebirth
   Room-based top-down shooter. Easy to get into, fun, and the right amount of frustrating for a rogue-like. 167 hours currently. There is still development going on for it as well, with an expansion planned for this year.
2. Nuclear Throne
   Top-down twinstick shooter (easiest played on mouse and keyboard however). Easy to get into, extremely difficult to "master". It's still in active development and improvements are being made by the developers
3. FTL: Faster Than Light
   Ship command survival game. Kind of on the opposite end of the rogue-like spectrum from Binding of Isaac. It's a decent game but it can get to you if you try too hard with it.

Isometric Hack & Slash -- These games are all relatively the same concept. If you're aware of the Diablo series, it's all the same general idea. Though don't think that I'm saying they're all the same, they play quite differently from one another.

1. Path of Exile
   This is a relatively modern game and has a high risk of being too much for your laptop, so be cautious on whether or not you want to try running it. It is, however, the best current example of the genre.
2. Torchlight, Torchlight II
   Two more great examples of the genre. As with most of the sequels on my list of games, be cautious on if you think the laptop can handle it.
3. Titan Quest
   And yet another good example of the genre. Support has largely dropped into a black hole for this game, so keep that in mind.
4. And obviously, Diablo 2+Diablo 2: Lord of Destruction
   I can't not recommend Diablo 2 out of this list. It'd be stupid if I didn't. The game is utterly fantastic, even years later. And better yet, your bad Internet gives you incentive to avoid Battle.net since it's a shithole for online play anyway! If you get some friends that want to play with you, it is a relatively low network demand game (for the modern era at least).