You have a few issues here. One is that you’re trying to mix legacy NFSS font families and modern fonts with fontspec. (This is possible, but more complicated than you need.) The second is that you want to format your chapter/section titles, but you haven’t loaded a package to do that (such as KOMA-Script’s scrlayer or titlesec). A third, trivial one is that you’re using even and odd pages without the twoside option to your document class.
Here’s a really ugly hack that makes only the minimal changes to your document:
\documentclass[twoside]{article}
\usepackage{fontspec} % I'm using XeLaTeX
\usepackage{fancyhdr}
\defaultfontfeatures{Scale=MatchLowercase}
\newfontfamily\DroidSerif{Droid Serif}
\pagestyle{fancy}
\fancyhead{}
\fancyhead[RO]{\textbf{\rightmark}}
\fancyfoot{}
\fancyfoot[LE,RO]{\fontfamily{\rmdefault}\thepage}
\begin{document}
\begin{titlepage}
\centering\DroidSerif
My title with droid font
\end{titlepage}
\section{\DroidSerif A section}
This text is in lmr style, but not the section title.
\end{document}
This is so basic, the section number is in a different font than the section title.
Here is a more full-featured solution using KOMA-Script:
\documentclass[headsepline, twoside]{scrartcl}
\usepackage{fontspec} % I'm using XeLaTeX
\usepackage{scrlayer-scrpage}
\defaultfontfeatures{Scale=MatchLowercase}
\newfontfamily\DroidSerif{Droid Serif}
\title{My Title with Droid Serif}
\author{Martin}
\pagestyle{scrheadings}
\lefoot{\pagemark}
\rofoot{\pagemark}
\automark[section]{section}
\automark*[subsection]{}
\addtokomafont{pagehead}{\DroidSerif\bfseries\upshape}
\addtokomafont{pagefoot}{\rmfamily}
\addtokomafont{title}{\DroidSerif}
\addtokomafont{section}{\DroidSerif}
\addtokomafont{subsection}{\DroidSerif}
\begin{document}
\begin{titlepage}
\maketitle
\end{titlepage}
\section{A section}
This text is in lmr style, but not the section title.
\newpage
Lorem ipsum dolor sit amet.
\end{document}
If you’re required to use PDFLaTeX, you need to remove \usepackage{fontspec} and add the appropriate NFSS commands. Load droid, and then lmodern to override it. The font family of Droid Serif is fdr. Although I don’t recommend this if you can use XeLaTeX, here’s a MWE:
\documentclass[headsepline, twoside]{scrartcl}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\usepackage[utf8]{inputenc} % The default since 2018
\usepackage{droid}
\usepackage{lmodern}
\usepackage{scrlayer-scrpage}
\newcommand\DroidSerif{\fontfamily{fdr}\selectfont}
\title{My Title with Droid Serif}
\author{Martin}
\pagestyle{scrheadings}
\lefoot{\pagemark}
\rofoot{\pagemark}
\automark[section]{section}
\automark*[subsection]{}
\addtokomafont{pagehead}{\DroidSerif\bfseries\upshape}
\addtokomafont{pagefoot}{\rmfamily}
\addtokomafont{title}{\DroidSerif}
\addtokomafont{section}{\DroidSerif}
\addtokomafont{subsection}{\DroidSerif}
\begin{document}
\begin{titlepage}
\maketitle
\end{titlepage}
\section{A section}
This text is in lmr style, but not the section title.
\newpage
Lorem ipsum dolor sit amet.
\end{document}
\newfontfamily\myfont{<name of font>}did not solved the problem at all, as this tutoral suggests (I'm using XeLaTeX). I've followed the steps to install the font manually using the instructions here and here (supposingly) without errors. Could you post a compilable code? – Martín Jun 08 '19 at 12:10