The style of an option does not change depending on the Expiration, as such, stocks are American Style and can be exercised at any time, whereas Index Options are still usually European Style and cannot be exercised until the expiry date.
Weekly Options can have far higher Gamma which causes the Delta of positions to change rapidly, especially as the Expiration date draws closer. Depending on when positions are initiated, they can be higher risk than Options with longer expirations, especially naked puts and calls where the capital loss is theoretically unlimited. In some cases, weekly Options can be used to fine-tune monthly positions and temporarily hedge or lock in profits (e.g. just prior to a news event).
Only CBOE Options for US stocks are currently offered with weekly expirations, most other exchanges for other regional Underlying assets or commodities are monthly only. CBOE recently announced the introduction of Wednesday Expiration Options which enables half-weekly expiry.
External LinksWeeklys Options
CBOE page describing weekly optionsCBOE to List SPX Wednesday-Expiring Weeklys Options
Press release from CBOE about Wednesday expiration weekly options.The Risks of Weekly Credit Spreads
Cautionary article from SteadyOptions.com about using weekly options for credit spread strategies (including Iron Condors).The Best Day to Sell Weekly Options
Josip Causic debates whether it is better to sell weekly options on Thursday when they are first made available to trade by CBOE or to wait until later on Friday to reduce risk and maximise the benefit of theta (time decay).