I have been working on a program for recording voice conversations as a mini project.
I have the recording working 100%, my problem is that I have no idea how to encode the wav file to a mp3 file.
Here is my recording and saving code:

procedure TForm1.Button1Click(Sender: TObject);
begin
  mciSendString('OPEN NEW TYPE WAVEAUDIO ALIAS mysound', nil, 0, Handle);
  mciSendString(
    'SET mysound TIME FORMAT MS BITSPERSAMPLE 16 CHANNELS 1 SAMPLESPERSEC 16384'
      + 'BYTESPERSEC 16384', nil, 0, Handle);
  mciSendString('RECORD mysound', nil, 0, Handle);
  Timer1.Enabled := true;
  sFormLabel := 'SED-Recorder : Recording';
  Form1.Caption := sFormLabel;
  StartTime := Now;
  iDisplayNumber := 0;
end;

procedure TForm1.Button2Click(Sender: TObject);
var
  sAudioName: string;
begin
  Timer1.Enabled := False;
  mciSendString('STOP mysound', nil, 0, Handle);
  SaveDialog1.Execute();
  sAudioName := SaveDialog1.FileName;
  mciSendString(PChar('SAVE mysound "' + sAudioName + '"'), nil, 0, Handle);
  mciSendString('CLOSE mysound', nil, 0, Handle);
  ProgressBar1.Position := 0;
  ProgressBar1.Visible := True;
  Saving.Enabled := True;
end;

That's a good question. But part of what many can't wrap their head around is that MP3 encoding was not intended to be free. Shocker?

Fraunhofer and how someone stole the encoder is at https://en.wikipedia.org/wiki/MP3#Going_public but the legal battles followed.

Today you might find it easier to encode it with a call to a LAME or LAME library. Example search.

Yes, I was going to suggest using the Lame mp3 encoding library. Also, there is open source code out there to do this, such as is used by the ffmpeg audio/video conversion tool.