Before I begin, I have to say that I loved the pun in Red and Blue where a TR member with a cold says Koffing evolves into Weezing. (And coughing does evolve into wheezing). And I couldn't stop laughing at how much Missingno. Master was flipping out. But anyways... Weezing is a classic Poison-type Pokemon (and the signature Pokemon of Team Rocket's James for the first five or so seasons of the anime) with an excellent base 120 defense. Its HP and Sp. Defense are both iffy, but get the job done. Attack and Sp. Attack are average at best, at base 90 and 85, respectively. Its Speed is a forgettable base 60, but that is faster than a number of other physical walls (like Forretress). Weezing's movepool and ability are also quite well suited for the physical walling role. The thing about Weezing is that its remained virtually unchanged. In such a dynamic environment, Weezing really struggles. Still, its good at what it does.
Recommended tiers: NU-UU
Levitate is Weezing's only ability. All Ground-type attacks except for Sand Attack have no effect on it. This is excellent, as it leaves Weezing with only one weakness: Psychic, and it makes Pokemon that rely on Earthquake as their main attack, like Gliscor and Hippowdon, weep.
Weezing has a pitiful at best physical movepool, so Special Attacks are the way to go. Sludge Bomb is a STAB option, though it leaves you helpless against Steels. Most Weezing run Flamethrower or Fire Blast instead. Thunderbolt, Shadow Ball, Dark Pulse, and Psybeam are additional special options.
Weezing's support movepool is where it's at. Pain Split is its best way to recover, and since Weezing has a rather low base 65 HP, it usually gains from the move. Will-O-Wisp is every physical attacker's nightmare, and Weezing uses it very well. Toxic is another status option, and if you want, you can combine it with Venoshock for great damage. Haze and Clear Smog serve the same purpose: removing stat changes. The difference is that Haze works on all Pokemon on the field while Clear Smog causes damage and only affects one target. Be very aware that Clear Smog is a Poison-type attack, which means Scizor and other Steels can boost without worry. Memento can be used if your Weezing looks like its about to go down, although its a bit too slow to really use it well. Destiny Bond has the same issue. Stockpile is good for stalling, but beware of crits.
Double/Triple options: An immunity to Earthquake is great, though Weezing can't damage an opponent much. A Weezing with Haze can partner well with a Pokemon that uses Draco Meteor, Overheat, or a similar stat-lowering move.
Countering Weezing: In OU, Heatran completely shuts Weezing down. In other tiers, Psychic type moves are the way to go. Pokemon with Magic Guard and Magic Bounce are particularly effective. Taunt shuts down any non-offensive Weezing. Try to use Special Attacks, as Weezing has a rather iffy base 70 Sp. Defense.