If you want a deployed static React App quickly:
S3, Terraform and create-react-app make it quite painless.

The following are instructions for creating a new React and making publicly addressable, using AWS and Terraform.

Install Libraries

Note: These are instructions for OSX
Note: I assume that you have an AWS account.
Note: I assume that you have npm installed.
Note: I assume that you have homebrew installed.

Install Terraform

brew install terraform

Install create-react-app

npm install -g create-react-app

Ensure your AWS credentials are in the right place

~/.aws/credentials

More info on AWS Security Credentials here

Build a new "create-react-app"

create-react-app my-app

Create S3 bucket and Policy

provider "aws" {
  region = "us-east-1"
}

variable "bucket_name" {
  default = "your-very-cool-bucket"
}

resource "aws_s3_bucket" "react_bucket" {
  bucket = "${var.bucket_name}"
  acl    = "public-read"

  policy = <<EOF
{
  "Id": "bucket_policy_site",
  "Version": "2012-10-17",
  "Statement": [
    {
      "Sid": "bucket_policy_site_main",
      "Action": [
        "s3:GetObject"
      ],
      "Effect": "Allow",
      "Resource": "arn:aws:s3:::${var.bucket_name}/*",
      "Principal": "*"
    }
  ]
}
EOF

  website {
    index_document = "index.html"
    error_document = "index.html"
  }
}

output "website_domain" {
  value = "${aws_s3_bucket.react_bucket.website_domain}"
}

output "website_endpoint" {
  value = "${aws_s3_bucket.react_bucket.website_endpoint}"
}

Apply your changes with terraform:

terraform apply

Place your code into the Bucket

Build your code:

npm run build

Push it up to S3:

aws s3 sync build s3://your-very-cool-bucket

Terrafuzz (AKA an easier way)

I also built a tool, that handles this setup for you.

Inside your React application:

git clone https://github.com/davidbegin/terrafuzz.git
cd terrafuzz
mv variables_template.tfvars variables.tfvars

# open up variables.tfvars and add your unique bucket name
# In our current example: your-very-cool-bucket

./terrafuzz.sh apply

and you are done, you now have a deployed website!
For more information visit Terrafuzz's Github