like this:

<TABLE cellpadding=3 cellspacing=0 width="100%"><TR><TD align="right" 
class="cartbasic"><P class="smallfade">First name</P></TD><TD 
class="cartbasic"><P>Peter</P></TD></TR>
<TR><TD align="right" class="cartbasic"><P class="smallfade">Last 
name</P></TD><TD class="cartbasic"><P>Smith</P></TD></TR>
<TR><TD align="right" class="cartbasic"><P 
class="smallfade">Address</P></TD><TD class="cartbasic"><P>Home street 
46</P></TD></TR>
<TR><TD align="right" class="cartbasic"><P 
class="smallfade">ZIP</P></TD><TD class="cartbasic"><P>87300</P></TD></TR>
</TABLE>

and the information taken out from there looks like this:

  First name  Peter
  Last name   Smith
  Address     Home street 46
  ZIP         87300

So, are there any obvious, simple HTML parse algorithms available for 
Delphi that could do this?

There are a lot of HTML-parsers in general. I have studied for instance 
the parsing code from this free Browser component:
http://pbear.com/htmlviewers.html.
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this parser. I do not get how to use this nice HTML-tool to parse data 
from my Html-tables. The code is far too advanced and complicated for my 
tiny need.

I also found that TWebBrowser component would be able to parse those 
HTML-tables, like this:

procedure TForm1.Button1Click(Sender: TObject);
var
  i, j: Integer;
  ovTable: OleVariant;
begin
  ovTable := WebBrowser1.OleObject.Document.all.tags('TABLE').item(0);
  for i := 0 to (ovTable.Rows.Length - 1) do
  begin
    for j := 0 to (ovTable.Rows.Item(i).Cells.Length - 1) do
    begin
      StringGrid1.Cells[j+1, i+1] :=
        ovTable.Rows.Item(i).Cells.Item(j).InnerText;
    end;
  end;

This way the data goes nicely go TStringGrid.