Weighted Pools

Overview
Weighted Pools are highly versatile and configurable pools. Weighted Pools use Weighted Math, which makes them great for general cases, including tokens that don't necessarily have any price correlation (ex. DAI/WETH). Unlike pools in other DeFi protocols that only provide 50/50 weightings, Balancer Weighted Pools enable users to build pools with different token counts and weightings, such as pools with 80/20 or 60/20/20 weightings.
Advantages
Exposure Control
Weighted Pools allow users to choose their levels of exposure to certain assets while still maintaining the ability to provide liquidity. The higher a token's weight in a pool, the less impermanent loss it will experience in the event of a price surge.
For example if a user wants to provide liquidity for WBTC and WETH, they can choose the weight that most aligns with their strategy. A pool more heavily favoring WBTC implies they expect bigger gains for WBTC, while a pool more heavily favoring WETH implies bigger gains for WETH. An evenly balanced pool is a good choice for assets that are expected to remain proportional in value in the long run.
Variable pool weight allows for fine-tuned exposure to assets
Impermanent Loss
Impermanent Loss is the difference in value between holding a set of assets and providing liquidity for those same assets.
Some people find the word "Impermanent" misleading and prefer to call it "Divergence Loss" or "Rebalancing Loss" because one token may perpetually out-value another token, and the loss may become... permanent.
For pools that heavily weight one token over another, there is far less impermanent loss, but this doesn't come for free; very asymmetric pools do have higher slippage when making trades due to the fact that one side has much less liquidity. 80/20 pools have emerged as a happy medium when balancing liquidity an Impermanent Loss mitigation.
Trading Pairs
Since each token in a pool can be traded with any other token in a pool, the number of trading pairs grows significantly with each additional token. By providing more trading pairs, pools are able to facilitate more swaps, giving them more opportunities to collect fees. 
The number of trading pairs in a pool follows the combinations equation nCr=n!r!(n−r)!_nC_r = \frac{n!}{r!(n-r)!}n​Cr​=r!(n−r)!n!​
. Where nnn
​ is 2 and rrr
​is the number of tokens in the pool.
Use Cases
Aave Safety Module
Aave is a decentralized non-custodial liquidity protocol where users can participate as depositors or borrowers. Aave uses pool tokens from the 80/20 AAVE/WETH Weighted Pool to lock funds in their Safety Module (SM) while still providing liquidity for the AAVE token.
The Safety Module solves the issues with traditional staking systems and market liquidity: Tokens with locking/reward schemes tend to suffer from low market liquidity and extreme volatility when high percentages of the total supply are being locked. With the ability of contributing to the SM not only by locking AAVE, but also by contributing with liquidity [on Balancer], stakers create a trustless and decentralized market with deep liquidity for trading AAVE against ETH.
Source: Aave Docs [Safety Module in Detail]
mStable Staking
mStable is a decentralized stablecoin protocol whose flagship product, mUSD, represents a basket of underlying stablecoins: DAI, sUSD, USDC and USDT. 
Similar to how Aave's Safety Module works, staked pool tokens from the 80/20 MTA/WETH Weighted Pool can be used in the mStable protocol as a backstop in case of a re-collateralization event to protect users from peg failures. By allowing users to stake pool tokens, mStable gains a more robust backstop and decreases the volatility of the MTA token.

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Balancer Pools

Stable Pools

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Overview

Advantages

Exposure Control

Trading Pairs

Use Cases

Aave Safety Module

mStable Staking