Dust An Elysian Tail

Side quests are secondary quests that, while not required, are often very rewarding both in items and money, but also XP. You can identify a side quest by the green leaf in your quest log. Look for this icon above an NPC to pick up new quests or complete/continue existing quests. You don't have to do these to finish the game, but in order to earn 3 achievements you're going to need to finish 20. This achievement requires only 10, but I've listed them all here for reference.

There are 21 total side quests so doing 20 will get you most of the optional quests. All of the side quests are located in either Aurora Village which you come to very soon or the subterranean village Mudpot which comes a bit later in the game. Unless mentioned all quests are available to start when you first meet the NPC.

NPC: Merchant
A New Venture

Aurora Village
The Blacksmith - NPC: Avgustin
Reunion - NPC: Moska
What's in the Box? - NPC: Reed
Out to Dry - NPC: Colleen
C-C-C-COMBO BREAKER - NPC: Corbin
Lost Time - NPC: Oneida (Must complete a boss fight before this is available)
Number One Fan - NPC: Moska (Must complete the quest Reunion before this is available)

For Fale's quests you need to gather the listed items. The best way to do this is to sell the items you find as soon as you get a new item, even if you just have one. What happens is whenever the game shows 'Stores Restocked' the quantity of each item for sale is increased in the merchant inventory. You can then go and buy these from any merchant. Hunting for the items is likely a waste of time as you can buy most of these pretty cheap.

NPC: Fale
Better than Nothing - Items Needed: 5 Imp Hides
Taking Point - Items Needed: 8 Beast Spears
Slimed - Items Needed: 4 Slimy Spikes & 4 Slimy Coats
Reinforcement - Items Needed: 2 Giant Rocks & 2 Giant Cores
Freedom of Movement - Items Needed: 6 Avee Wings & 6 Avee Claws
Find the Sparklies - Items Needed: 3 Florn Tentacles & 3 Florn Sparks

Mudpot
A Better Fidget - NPC: SmoBop
Red Moss - NPC: FloHop
Over the Fence - NPC: Gappy
Hittin' Stick - NPC: Blop
Trolk Vengeance - NPC: Blop (Must complete the quest Hittin' Stick before this is available)
Snowflakes - NPC: Bopo
Pearly Whites - NPC: Blop (Must complete a boss fight before this is available)