curl -u username:password ...

$securePwd = ConvertTo-SecureString "password" -AsPlainText -Force
$credential = New-Object System.Management.Automation.PSCredential ($username, $securePwd)

Invoke-RestMethod -Credential $credential ...

$base64AuthInfo = [Convert]::ToBase64String([Text.Encoding]::ASCII.GetBytes(("{0}:{1}" -f $username,$password)))

Invoke-RestMethod -Headers @{Authorization=("Basic {0}" -f $base64AuthInfo)} ...

$pwd = ConvertTo-SecureString "MyPassword" -AsPlainText -Force
$cred = New-Object Management.Automation.PSCredential ('PsUser', $pwd)

Invoke-RestMethod 'http://httpbin.org/basic-auth/PsUser/MyPassword' -cred $cred

$username = "username";
$password = ConvertTo-SecureString –String "password" –AsPlainText -Force
$credential = New-Object –TypeName "System.Management.Automation.PSCredential" –ArgumentList $username, $password

$base64AuthInfo = [Convert]::ToBase64String([Text.Encoding]::ASCII.GetBytes(("{0}:{1}" -f $username,$password)))

$getProjectUri = "yourUri"
Invoke-RestMethod -Method Get -Uri $getProjectUri -Headers @{Authorization = "Basic $base64AuthInfo" } -Credential $credential -ContentType "application/json"

$creds = Get-Credential
$unsecureCreds = $creds.GetNetworkCredential()
$base64AuthInfo = [Convert]::ToBase64String([Text.Encoding]::ASCII.GetBytes(("{0}:{1}" -f $unsecureCreds.UserName,$unsecureCreds.Password)))
Remove-Variable unsecureCreds

Invoke-RestMethod -Headers @{Authorization=("Basic {0}" -f $base64AuthInfo)} ...