Konnichiwa! Similar to my 3DS Pokémon Game recommendations, another Friend asked me about the Pokémon Games for the DS/DSi, and so here they are!

Diamond/Pearl

The first Pokémon Games for the DS, the first Gen 4 Games, and the first "modern" Pokémon Games (meaning if you're used to playing these, it won't be that difficult to adjust to any of the newest Games, other than the some of controls).
When these Games first came out, they were revolutionary.

Not only did they completely change the way Physical and Special moves work (which had a MASSIVE impact on Battling), they also were the first Games that allowed online Battling and Trading (which AFAIK don't work anymore).

These Games also have way more Pokémon in them than any of the older Games, although most aren't available until you complete the Game.

Key Features include:

All Gen 4 Pokémon (between both Games, and excluding Giratina Origin Forme)

The Pokétch

A cool in-game smartwatch, with various Apps, such as the Dot Artist, the Calculator, and the Type Checker.

The Underground

A Massive, Underground World, where you can dig for Treasure (such as Fossils), have a Secret Base (which you can customize), and play with others nearby.

The Poké Radar

Lets you find rare Pokémon, and even makes it easier to find Shiny ones.

Pal Park

A place for moving in Pokémon from the GBA Games.

Game Corner/Safari Game

The Game Corners are really cool in these Games, and neither Game Corners, nor the Safari Game even exist in Gen 5 and beyond.

Platinum

An enhanced version of D/P, with a new Story, both of the main Legendaries (and even a new one: Giratina Origin Forme), as well as some new Features, including:

The Battle Frontier

They brought back the Battle Frontier, a place with many Battling Facilities.
(In D/P, they only have the Battle Tower.)

Shard Moves

They added many Move Tutors, who can teach many cool moves (including one of my favorites: Outrage) in exchange for different colored Shards.

New Pokédex

They expanded the Pokédex, allowing you to get more Pokémon, before you complete the Game.

Heart Gold/Soul Silver

My 2nd favorite Pokémon Games of all time.
If you played the original Games for the GBC, you'll love these.
They have most of the older Pokémon that were left out of D/P/Pl in them, and have many Gen 3/4 Pokémon as well.
These Games (along with the original Gen 2 Games) are the only Pokémon Games with 2 different Regions (Johto and Kanto) in 1 Game.

Key Features include:

The Pokéwalker

A cool pedometer that allows you to (literally) walk with your Pokémon IRL.
Also, you can catch rare Pokémon with it, or find rare Items.

Walking with your Pokémon

You can walk with your Pokémon in-game, too.
Your lead Pokémon stays outside it's Poké Ball, like Ash and his Pikachu.

The Pokéathlon

A bunch of fun, Olympic-like Mini-Games, played with 3 of your Pokémon.

Game Corner

The Game Corners have an interesting Card Game, that tests both your math skills, and your intuition, instead of Slot Machines like usual.

Custom Safari Zones

You can design your own Safari Zone, with different Pokémon in it, depending upon your design.

Special Pokémon

They brought back both the Red Gyarados, and the Extreme Speed Dratini.

Black/White

The first Gen 5 Games.
They did many odd things in these Games, like starting out on a completely new "Route 1", and not having ANY older Pokémon in them, until you complete the Game.

They also added a TON of online features. Unfortunately, they no longer function (AFAIK), and the Games were heavily dependent on them (such as for growing Berries).

They are still really cool Games, however, and have a TON of brand new Pokémon.

Features include:

Triple Battles

Battling with 3 Pokémon out at a time.
This is my favorite kind of Battle.

Pokémon Musicals

You can dress up your Pokémon, and put them in Musicals.

The C-Gear

A neat device that has many different communication features.

The Poké Transfer Lab

A place for moving in Pokémon from the Gen 4 Games.
(Requires 2 Systems)

Black 2/White 2

The sequels to B/W. If you didn't play those, you may not get certain references, but the Games will still be fun.
These Games are far less Internet-dependent than B/W, and have new features, including:

The PWT

IMO, the single best Battling Facility in ANY Game.
You can Battle many different Gym Leaders and Champions from older Games, in Single, Double, or Triple Battles.

Join Avenue

A cool place that can be customized with different shops, that provide goods and services.

Pokéstar Studios

A place for filming Movies with your Pokémon.

Conclusion:

If you like Pokémon Contests a lot, want Giratina Origin Forme (in Gen 4), or just like the various features, go with Platinum.

If you're nostalgic, or like the whole "walk with your Pokémon" thing, or just like Dragons (all of which apply to me), then go with HG/SS.

If you want the newest Games you can get, or just really like Battling, go with B2/W2.