Monthly achievements are similar to daily achievements but they require more time and effort to complete. Monthly achievements change form month to month and you have to complete all of them during the specific month to count.
There are four different types of achievements each with 4 tiers. Augusts monthly achievements were as follows:
Monthly WvW Player Kills – (4 tiers) – 5 to 50 Invaders killed
Monthly Event Participation – (4 tiers) - 25 to 100 dynamic events completed
Monthly Salvage Kit Usages – (4 tiers) – 125 to 500 items salvaged
Experience Survivor – (4 tiers) – 25,000 xp to 100,000 experience without Death
Upon completing all of the monthly achievements you will get a nice chest with gold, experience and other useful rewards.
Monthly achievements are account wide, so you can do them on all of your characters, but only one will get the rewards. Also the same tips that you can use for daily achievements apply also for monthly achievements so check them out here.