The Daily Grind is a series of tasks which can be completed to earn special rewards.
To access the Daily Grind, you must click the Daily Grind button in the upper right of the Kingdom screen. From there you can claim the prizes for the objectives you've completed.
Although they are worded as days, you do not have a time limit to complete them (plus, it would be really hard to meet the deadlines), nor do you need to do them in order.
The Daily Grind is available until it's expired, and then disappears forever. However there is time limit of 14 days - this limit only applies to you and is not linked to server time. Every "Day" requirement is doable, just keep upgrading to avoid being stuck by cooldowns. If you are not lucky to get a level 3 jewel anywhere then make explorations at the Jewel Crafter to get jewels and then Forge those.
Day 4 and Day 7 should be your goal to finish.