The National - The Alligator
Broken Social Scene - Broken Social Scene
The Antlers - Hospice
Modest Mouse - The Moon & Antarctica
Owen Pallet - Heartland

1. The Replacements - Let It Be
2. Violent Femmes - Violent Femmes
3. Sonic Youth - Daydream Nation
4. Minutemen - Double Nickels on the Dime
5. The White Stripes - White Blood Cells

This is a very hard one... Im going to think on it. Is this it by the Strokes feels like it has to be on here.

Gee that's hard. Indie is such a deceptively large genre. For a noob i'd say

Stone Roses -S/t
Pixies - Doolittle
Jesus And Mary Chain - Pyschocandy
The La's - s/t
Blur - Parklife

All predictable yet accessible choices.

I'd say:

The Smiths - s/t
Arcade Fire - Funeral
Radiohead - Amnesiac or OK
Belle & Sebastian - Tigermilk/DCW/If you're feeling sinister
NMH - An Aereoplane over the sea (but that's so obvious)
The Postal Service - Sleeping in (Has to be one of the albums I have listened to most in my life)

Olivia Tremor Control - Dusk at Cubist's Castle
Guided by Voices - Alien Lanes
Sonic Youth - Confusion is Sex
The Smiths - The Smiths
My Bloody Valentine - Loveless

Neutral Milk Hotel - In the Aeroplane Over the Sea
Elliott Smith - Either/Or
The National - Alligator
Pavement - Crooked Rain, Crooked Rain
Modest Mouse - The Moon and Antarctica

This was a tough one. So many great albums to choose from. For the moment, here's my 5.

Replacements - Let It Be
Sonic Youth - Daydream Nation
Pavement - Crooked Rain, Crooked Rain
Sebadoh - Bake Sale
MinuteMen - Double Nickels On The Dime

I'm sure I'll want to edit this list before too long. It was hard not to include:

Guided By Voices - Bee Thousand
Elliott Smith - Either/Or
Dinosaur Jr - You're Living All Over Me
Archers Of Loaf - Icky Mettle
Husker Du - New Day Rising

and so many more...

In no particular order, but I am still pretty sure that these are the five I'd include:

Bonnie 'Prince' Billy - I See A Darkness
Elliott Smith - Either/Or
Bright Eyes - Lifted, Or The Story Is In the Soil
Neutral Milk Hotel - In the Aeroplane Over the Sea
The Smiths - The Smiths

Very 'safe' (in listing well known albums) lists. I suppose when someone says essential people think it must be famous singers or groups who have received a lot of praise generally.