Which kind of statistical test should I try for a sample with bivariate data? I'm trying to analyze data from a questionnaire, showing graphical results. I used dispersion charts for correlating bivariate data with continuous and discrete ranges, showing also a trend line on the distribution. What about correlating binary data samples and discrete samples? E.g., I have the following 2 series: 
X = (1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0) 
Y = (4, 4, 2, 4, 3, 4, 1, 4, 3, 3, 3, 3, 3, 3)

The first variable may have only values 0 or 1, the second one may have values in a range [1,4]. A friend suggested that t-test, Chi-squared and Fisher's test could be interesting. But I can't imagine how they can be used in a graphical way.
 A: The question asks for ways to display bivariate discrete data using Excel.  Although that software is notoriously limited in graphical capabilities, it is still able to generate many different useful graphics.  Let's use the example to illustrate.  Here are the data, laid out as they might be in a spreadsheet:
X, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0
Y, 4, 4, 2, 4, 3, 4, 1, 4, 3, 3, 3, 3, 3, 3

Scatterplot
By default, a good way to display bivariate data is with a scatterplot: distance along one coordinate corresponds to X, distance along the other coordinate to Y, and a standard symbol is drawn at the Cartesian location (X,Y) (one symbol for each (X,Y) pair).
The problem with discrete data is that many points may coincide on the scatterplot.  For instance, the example data would show seven points stacked at the location (0,3).  A solution is to jitter the points: move them randomly a little bit.  Do this by adding a small multiple of RAND() - 1/2 to the values.  Here, I chose 1/3 for the multiple because it disaggregates the stacked points but keeps them visually clustered near their correct locations:

Bubbleplot
When jittering might confuse the audience, or when many points might be included in clusters, it may help to summarize the count of points at each location (X,Y) with a "bubble plot."  This one is annotated with the count, because people usually do not compare areas of circles correctly:

Producing this required creating a table of (X, Y, Count) triples: do this using COUNTIF.  For this example, the X values were originally placed in cells A2:A15 and the Y values next to them in cells B2:B15.  To obtain the table, I computed
10*A2 + B2

in cell C2 and copied it down to cell C15, creating a "code" column.  Its values are
14  14  2   14  3   4   1   14  3   3   3   3   3   3

They could then be counted after setting up parallel columns of all possible X and Y values:
X, Y
0, 1
0, 2
...
1, 4

I put these into columns M and N.  To the right of each entry the count is computed with a formula like
=COUNTIF($C$2:$C$15, 10*M2+N2)

In effect, each entry is converted into a unique numeric code with 10*M2+N2 and then that code is looked up in the "Code" column and counted.  (One begins to understand why databases and statistical software are usually used for such visualizations rather than spreadsheets.)
Point plot
We can "slice" the data by conditioning on one value, such as the binary X value.  For X=0 this produces a sequence of (Y, Count) pairs and for X=1 it produces another sequence of (Y, Count) pairs: use COUNTIF to compute these, laying them out in a rectangular array like this:
,  1, 2, 3, 4
0, 1, 1, 7, 1
1, 0, 0, 0, 4

The rows provide the X values (labeled by the left column) and the columns provide the Y values (labeled by the top row).  The table entries are the counts.
Parallel point plots display these values:

Note the need for a legend and to distinguish the values of X by means of visually different symbols (solid dark circles and hollow light squares here).
Point plots provide accurately readable displays of the counts: people can compare the heights quickly and easily.  The faint grid makes reading off the counts more reliable without intruding on the visual display.
Bar chart
Bars can "lie" when they are based at arbitrary values, but when they are based at zero, their lengths accurately reflect the values they are intended to depict.  As such, we can use bars instead of points:

In some applications care is needed to distinguish zeros (shown as the absence of a bar) from missing values (also shown as the absence of a bar).  That's not necessary in this example, because a missing count for an (X,Y) pair is not possible.
Pseudo 3D chart
3D representations of data are difficult to interpret and read quantitatively, but with certain complex datasets they can sometimes be useful.  In this instance, we erect a 3D bar at each location (X,Y) whose pseudo-height is directly proportional to the count:

Quickly, now: does the tallest bar have a height of 6, 7, or 8?  Imagine trying to figure this out when there are many possible values of X and Y in the image.  Usually, such charts are reserved for annual reports and other slick, glossy materials intended for propaganda instead of presenting information.
A: If you want to correlate X and Y, then you could consider point biserial correlation:
$R_{pb} = \frac{M_1 - M_0}{s_n}\frac{n_1n_0}{n*2}$ where $s_n$ is the standard deviation of (in your case) Y and the subscripts refer to the two groups as defined by X, if you are willing to posit that Y is "really" continuous (e.g. it's a marker on some latent scale).
If you want to see if Y is different for the two levels of X, you could do a t-test.
If you want to see if there is association between X and Y, then you could consider either chi-square or Fisher's exact test; but if you are willing to assume that Y is ordinal, you could consider a test that accounts for that, such as the Jonckheere Terpstra test. 
For plotting in Excel, @whuber already gave several ideas; if you do a lot of plotting and insist on Excel, you should read Jon Peltier's blog he does amazing things, given Excel's limits.
A: This is a very standard analysis. A chi-square test or Fisher's exact test would indeed be valid approaches. I don't understand what you mean by asking how to imagine their use in a graphical way, but one approach would be to produce a proportions chart. Here, the two X levels would be depicted on the horizontal axis with groups 0 and 1 and the proportions in each group responding in each polytomous Y category would be shown in a stacked format, where the vertical axis is in units of percentages. One would imagine that, without association, these bars would look the same in both X groups.
