Delete an element from my ListView

To make the application more readable, I suggest that you create an object of type Host.

public class Host {

    private String mNome;
    private String mDescricao;
    private String mStatus;

    private void setNome(String nome) {
        mNome = nome;
    }

    private String getNome() {
        return mNome;
    }

    private void setDescricao(String descricao) {
        mDescricao = descricao;
    }

    private String getDescricao() {
        return mDescricao;
    }

    private void setStatus(String status) {
        mStatus = status;
    }

    private String getStatus() {
        return mStatus;
    }
}

Then you will modify your Adapter to receive a list of objects with type Host.

public class ListViewAdpterInicial extends BaseAdapter {

    Context context;
    List<Host> hostsList;

    public ListViewAdpterInicial(Context context) {
        this.context = context;
    }

    public void setItems(List<Host> hostsAux) {
        hostsList = hostsAux;
    }

    @Override
    public int getCount() {
        return hostsList.length;
    }

    @Override
    public Object getItem(int position) {
        return null;
    }

    @Override
    public long getItemId(int position) {
        return 0;
    }

    public View getView(int position, View convertView, ViewGroup parent) {
        TextView txtHost;
        TextView txtDescricao;
        TextView txtStatus;

        LayoutInflater inflater = (LayoutInflater)context.getSystemService(Context.LAYOUT_INFLATER_SERVICE);
        View itemView = inflater.inflate(R.layout.texto_inicial_host, parent, false);

        txtHost = (TextView) itemView.findViewById(R.id.txt_host_inicial);
        txtDescricao = (TextView) itemView.findViewById(R.id.txt_host_descricao);
        txtStatus = (TextView)itemView.findViewById(R.id.txt_host_status);

        Host host = hostsList.get(position);

        txtHost.setText(host.getName());
        txtDescricao.setText(host.getDescricao());
        txtStatus.setText(host.getStatus());

        return itemView;
    }
}

Then in your class, you generate a list (just an example) of the type of that object and pass it to the Adapter.

List<Host> hostsList = new ArrayList<Host>();

Host host = new Host();
host.setNome("Host1");
host.setDescricao("Estado do Host");
host.setStatus("DOWN");

hostsList.add(host);

host.setNome("Host2");
host.setDescricao("Estado do Host");
host.setStatus("UNREACHABLE");

hostsList.add(host);

host.setNome("Host3");
host.setDescricao("Host");
host.setStatus("DOWN");

hostsList.add(host);

adapter = new ListViewAdpterInicial(getActivity().getBaseContext());
list.setAdapter(adapter);

adapter.setItems(hostsList);

Finally you should let your Adapter know you have new items and it should update itself.

adapter.notifyDataSetChanged();

When you want to update the Adapter list, simply generate a new one (with only the 2 items you want), do the setItems again and then the notifyDataSetChanged;