I have two issues:
I want to change the numeration in section, now I have:

but I would like to have:
10 Charakteryzacje zmiennej losowej
- Definition
- Theorem
It means that the first number which indicates the section number is deleted.
- Second issue is connected with the numeration too. Now I have that the numeration of all the equations spreads from 1 to ... in the entire document. I want it to start from 1 in each new section.
Edit: Inside my document I have:
\newtheorem{tw}{Theorem}[section]
\renewcommand{\theorem}{\arabic{tw}}
\newtheorem{stw}[tw]{Stwierdzenie}
\newtheorem{fakt}[tw]{Fakt}
\newtheorem{lemat}[tw]{Lemat}
\theoremstyle{definition}
\newtheorem{df}[tw]{Definicja}
\newtheorem{ex}[tw]{Przykład}
\newtheorem{uw}[tw]{Uwaga}
\newtheorem{at}[tw]{Wniosek}
and it doesn't work in a correct way, below is the class I use:
\documentclass[12pt, a4paper]{article}
\usepackage{marginnote}
\usepackage[top=2.5cm, bottom=3.5cm, outer=2cm, inner=2cm, heightrounded, marginparwidth=2.5cm, marginparsep=2cm]{geometry}
\usepackage[polish]{babel}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[pdftex,linkbordercolor={0 0.9 1}]{hyperref}
\usepackage{amsthm,amsmath,amsfonts,amssymb,mathtools}
\usepackage{url}
\usepackage{enumerate}
\usepackage{graphicx}
\usepackage{bbm}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhf{}
\fancyfoot[R]{\textbf{\thepage}}
\fancyhead[L]{\small\sffamily \nouppercase{\leftmark}}
\renewcommand{\headrulewidth}{0.4pt}
\renewcommand{\footrulewidth}{0.4pt}
\usepackage{chngcntr}
\counterwithin*{equation}{section}
\newtheorem{tw}{Theorem}[section]
\renewcommand{\theorem}{\arabic{tw}}
\newtheorem{stw}[tw]{Stwierdzenie}
\newtheorem{fakt}[tw]{Fakt}
\newtheorem{lemat}[tw]{Lemat}
\theoremstyle{definition}
\newtheorem{df}[tw]{Definicja}
\newtheorem{ex}[tw]{Przykład}
\newtheorem{uw}[tw]{Uwaga}
\newtheorem{at}[tw]{Wniosek}
\begin{document}
\end{document}

\newtheorem{mytheorem}{Theorem}in your document, add a line\renewcommand{\themytheorem}{\arabic{mytheorem}}. – karlkoeller Aug 10 '13 at 17:51\documentclass{...}and ending with\end{document}. – karlkoeller Aug 10 '13 at 17:52\renewcommand{\thetheorem}not\renewcommand{\theorem}– Andrew Swann Aug 10 '13 at 18:44Theorem~\ref{thm:a} in Section~\ref{sec:b}) instead of the simpler “Theorem 10.2”. – egreg Oct 12 '13 at 16:00